A speedometer answers a question that a whole-trip average cannot: what is happening now? Calculus gives us a controlled way to move between that local view and the accumulated journey.
This chapter builds the connection from small changes and small accumulated contributions. We use polynomial models and simple graphs before moving to chain rules, varying forces and sampled data. No previous calculus notation is assumed, but you should be comfortable with signed quantities, powers and basic motion.
What you will learn to do
- Interpret a derivative as a local rate and a tangent slope.
- Differentiate simple models while retaining their physical units.
- Apply the chain rule to a changing inner quantity.
- Interpret an integral as signed accumulation and include its initial condition.
- Separate distance from displacement and work from impulse.
- Distinguish exact model calculations from estimates based on sampled data.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics Learning goals: - Interpret a derivative as a local rate and a tangent slope. - Differentiate simple models while retaining their physical units. - Apply the chain rule to a changing inner quantity. - Interpret an integral as signed accumulation and include its initial condition. - Separate distance from displacement and work from impulse. - Distinguish exact model calculations from estimates based on sampled data. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
How does a secant slope on a position graph differ from a tangent slope?
Compare your reasoning
A secant compares two separated times and gives an average velocity. A tangent is the limiting local slope and gives instantaneous velocity. Both retain units of length per time.
Revisit the explanation →01 / Rates and accumulation: calculus for physics
From a change over an interval to a rate at an instant
An average velocity answers how much position changed over a nonzero time interval. An instantaneous velocity asks for the rate at one instant. To connect them, compare the position at time t with the position a short time h later, divide by h, and investigate what happens as h becomes smaller.
The derivative is the limiting value of that ratio when the limit exists. We are not dividing a zero displacement by a zero time. For each nonzero h the ratio is defined; the limit describes the value approached by a sequence of such ratios. Geometrically, secant lines through two points approach the tangent at the selected point.
The same reasoning applies beyond motion. Current is the rate at which charge passes a cross-section. Power is an energy-transfer rate. A derivative always needs its independent variable: a change of energy per time is power, whereas a change of potential energy per position is related to force.
If f has units of metres and t has units of seconds, df/dt has units of m/s. Another derivative with respect to time produces m/s².
WORKED EXAMPLE
Watch an average rate approach an instantaneous one
A model has x = t² metres when t is in seconds. Find the instantaneous velocity at t = 3 s from the difference quotient. The coefficient multiplying t² carries units m/s².
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics
Focus: From a change over an interval to a rate at an instant
Learning goals:
- Interpret a derivative as a local rate and a tangent slope.
- Differentiate simple models while retaining their physical units.
- Apply the chain rule to a changing inner quantity.
- Interpret an integral as signed accumulation and include its initial condition.
- Separate distance from displacement and work from impulse.
- Distinguish exact model calculations from estimates based on sampled data.
Equations in this section (LaTeX):
\frac{df}{dt}=\lim_{h\to0}\frac{f(t+h)-f(t)}{h},\qquad v=\frac{dx}{dt},\qquad a=\frac{dv}{dt}
Meaning: If f has units of metres and t has units of seconds, df/dt has units of m/s. Another derivative with respect to time produces m/s².
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Rates and accumulation: calculus for physics
Build useful differentiation rules from that meaning
For a constant f, the numerator f(t + h) − f(t) is zero, so the derivative is zero. For f = At, the numerator is Ah and the derivative is A. For a square, expansion gives the pattern demonstrated above. The power rule generalises the result: differentiating tⁿ gives ntⁿ⁻¹ wherever the function is differentiable.
A constant coefficient stays as a multiplier. Derivatives distribute across sums because the changes of separate terms add. These two rules let you differentiate a polynomial term by term. Keep the coefficient’s physical units: in x = x₀ + ut + ½at², x₀ is a length, u is a velocity and a is an acceleration.
The coefficient n in the power rule is a dimensionless exponent; it is not a new physical quantity. The independent variable must remain clear. Differentiating with respect to position, time or temperature describes different rates and produces different units.
| Function of t | Derivative with respect to t | Reason or condition |
|---|---|---|
| C | 0 | C is constant with respect to t |
| Atⁿ | nAtⁿ⁻¹ | A is constant; use where differentiable |
| f(t) + g(t) | f′(t) + g′(t) | Changes add |
| sin(ωt) | ω cos(ωt) | The argument is in radians; ω is constant |
| exp(kt) | k exp(kt) | kt is dimensionless; k is constant |
WORKED EXAMPLE
Recover motion from a position model
A particle follows x(t) = 4 + 3t + 0.5t², with x in metres and t in seconds. Find its position, velocity and acceleration at t = 2 s.
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Velocity is the slope, not the position
- Velocity
- 3
STOP & THINK
A position model is x = t³ − 3t metres, with t in seconds and coefficients in compatible SI units. What is the velocity at t = 2 s?
Answer in m/s.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics Focus: Build useful differentiation rules from that meaning Learning goals: - Interpret a derivative as a local rate and a tangent slope. - Differentiate simple models while retaining their physical units. - Apply the chain rule to a changing inner quantity. - Interpret an integral as signed accumulation and include its initial condition. - Separate distance from displacement and work from impulse. - Distinguish exact model calculations from estimates based on sampled data. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Rates and accumulation: calculus for physics
A changing inner quantity introduces another rate
Suppose y depends on a quantity z, and z itself changes with time. A small time change first changes z and then changes y. The total rate is the rate of y per z multiplied by the rate of z per time. This is the chain rule, and its units cancel exactly as that explanation suggests.
For an oscillation sin(ωt), the inner phase changes at rate ω. Differentiating the sine without multiplying by ω would omit how rapidly the phase advances. If the input is t/τ, its derivative is 1/τ. This extra factor is also required by the units.
The chain rule does not mean every bracket produces a mysterious new formula. Name the inner quantity, differentiate the outer dependence with respect to it, and then multiply by its actual rate. This two-stage approach is useful in both algebraic and trigonometric physics models.
For a length y and a dimensionless z, the first factor has units of length and the second of inverse time, giving velocity.
WORKED EXAMPLE
Do not lose the clock inside the bracket
A position is x = 5(1 + 2t)² metres, with t in seconds. The coefficient 2 has units s⁻¹. Find the velocity at t = 0.5 s.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics
Focus: A changing inner quantity introduces another rate
Learning goals:
- Interpret a derivative as a local rate and a tangent slope.
- Differentiate simple models while retaining their physical units.
- Apply the chain rule to a changing inner quantity.
- Interpret an integral as signed accumulation and include its initial condition.
- Separate distance from displacement and work from impulse.
- Distinguish exact model calculations from estimates based on sampled data.
Equations in this section (LaTeX):
\frac{dy}{dt}=\frac{dy}{dz}\frac{dz}{dt}
Meaning: For a length y and a dimensionless z, the first factor has units of length and the second of inverse time, giving velocity.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.04 / Rates and accumulation: calculus for physics
Reverse the question: a rate builds a total change
If velocity is approximately constant over a short time interval Δt, the displacement is approximately vΔt. Divide a longer interval into many short pieces, calculate those contributions and add them. As the pieces become arbitrarily small under suitable regularity conditions, their sum becomes a definite integral.
The integral sign is an instruction to accumulate. Its limits identify where the accumulation starts and ends. The symbol dt indicates that the narrow widths are time intervals. The integrand v(t) supplies the corresponding rate. Together, v dt has units of length.
An integral gives a change when the integrand is a rate of change. To obtain the final position, add the initial position. The initial condition is not optional information: two particles can have identical velocities at all times while remaining separated by a fixed distance.
The sum samples each small interval. The limiting construction is a signed area under the velocity–time curve, not a sum of speeds unless speed is explicitly used.
Likewise, integrating acceleration over time gives a change in velocity, integrating current gives transferred charge, and integrating a force component over time gives impulse. Integrating force over displacement instead gives work. The integration variable matters as much as the quantity being integrated.
STOP & THINK
What does the time integral of acceleration determine directly?
- Final velocity without any initial condition
- Change in velocity
- Distance travelled
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics
Focus: Reverse the question: a rate builds a total change
Learning goals:
- Interpret a derivative as a local rate and a tangent slope.
- Differentiate simple models while retaining their physical units.
- Apply the chain rule to a changing inner quantity.
- Interpret an integral as signed accumulation and include its initial condition.
- Separate distance from displacement and work from impulse.
- Distinguish exact model calculations from estimates based on sampled data.
Equations in this section (LaTeX):
x(t_2)-x(t_1)=\lim\sum v(t_i)\,\Delta t_i=\int_{t_1}^{t_2}v(t)\,dt
Meaning: The sum samples each small interval. The limiting construction is a signed area under the velocity–time curve, not a sum of speeds unless speed is explicitly used.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.05 / Rates and accumulation: calculus for physics
Use an antiderivative, then apply the endpoints
An antiderivative of f is a function whose derivative is f. For example, t³/3 has derivative t². The fundamental theorem of calculus allows a definite integral of a continuous function to be evaluated by subtracting its antiderivative at the two endpoints.
There are infinitely many antiderivatives differing by an additive constant, because differentiating a constant gives zero. An indefinite integral therefore includes a constant of integration. For a definite integral, the same constant appears at both endpoints and cancels. When reconstructing a physical state, the initial condition fixes the otherwise unknown constant.
The power rule for integration raises the exponent by one and divides by that new exponent, except for exponent −1. The integral of 1/t is ln|t| plus a constant on an interval avoiding zero. In physical logarithmic expressions, combine quantities into dimensionless ratios before taking logarithms.
The endpoint subtraction evaluates a net accumulation. Reversing the integration limits reverses its sign.
WORKED EXAMPLE
Recover velocity and position from a varying acceleration
A particle has a(t) = 2t m/s² when t is measured in seconds; the proportionality coefficient has units m/s³. At t = 0 its velocity is 1 m/s and its position is zero. Find v and x at t = 2 s.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics
Focus: Use an antiderivative, then apply the endpoints
Learning goals:
- Interpret a derivative as a local rate and a tangent slope.
- Differentiate simple models while retaining their physical units.
- Apply the chain rule to a changing inner quantity.
- Interpret an integral as signed accumulation and include its initial condition.
- Separate distance from displacement and work from impulse.
- Distinguish exact model calculations from estimates based on sampled data.
Equations in this section (LaTeX):
\int_{a}^{b}f(t)\,dt=F(b)-F(a),\quad F^{\prime}(t)=f(t);\qquad\int t^n\,dt=\frac{t^{n+1}}{n+1}+C\quad(n\ne-1)
Meaning: The endpoint subtraction evaluates a net accumulation. Reversing the integration limits reverses its sign.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.06 / Rates and accumulation: calculus for physics
Cancellation in an integral has a physical meaning
A velocity can be positive during one part of a journey and negative during another. Its signed integral gives net displacement. Portions below the zero line subtract from portions above it. The integral of speed, |v|, instead counts all travelled length positively.
To calculate a distance when velocity changes sign, first find the zero crossings in the time interval. Split the integral there, determine the sign on each subinterval and add the magnitudes of the displacement contributions. Taking the absolute value only after integrating velocity generally gives the magnitude of the net displacement, which can be much smaller than the distance.
The distinction also appears in electric current and energy transfer. A signed charge transfer through a chosen surface can cancel between opposite directions. A net work integral can be zero even though positive and negative energy transfers occurred during different parts of a motion.
WORKED EXAMPLE
A zero displacement with a nonzero journey
A particle has v(t) = 4 − 2t m/s on the interval 0 ≤ t ≤ 4 s. Find its displacement and total distance.
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STOP & THINK
A velocity is v(t) = 3 − t m/s for 0 ≤ t ≤ 6 s. What total distance is travelled?
Answer in m.
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics Focus: Cancellation in an integral has a physical meaning Learning goals: - Interpret a derivative as a local rate and a tangent slope. - Differentiate simple models while retaining their physical units. - Apply the chain rule to a changing inner quantity. - Interpret an integral as signed accumulation and include its initial condition. - Separate distance from displacement and work from impulse. - Distinguish exact model calculations from estimates based on sampled data. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Rates and accumulation: calculus for physics
Choose the variable from the transfer you are measuring
Work by a force along a straight path accumulates the component of force along that path multiplied by displacement. For a variable force, a single final force value does not represent the whole journey. Integrate the force component over position, or calculate the corresponding area if its shape is simple.
Impulse instead accumulates force over time. Work and impulse have different units and answer different physical questions. Work changes kinetic energy for the chosen particle model; impulse changes momentum. The same force can produce different values of these integrals because displacement and elapsed time need not have the same history.
In more than one dimension, work uses the scalar product of force and the small displacement. A perpendicular force contributes no work at that instant even if it changes the direction of motion. This directional condition is essential when extending an area calculation beyond a one-dimensional graph.
Work has units N·m = J; impulse has units N·s = kg·m/s. The integration variable makes the difference.
WORKED EXAMPLE
Accumulate a growing force over position
A 3 kg particle starts from rest and moves along +x under the only horizontal force Fₓ = (6 N/m)x. It moves from x = 1 m to x = 2 m on a smooth horizontal track. Find the work and final speed.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics
Focus: Choose the variable from the transfer you are measuring
Learning goals:
- Interpret a derivative as a local rate and a tangent slope.
- Differentiate simple models while retaining their physical units.
- Apply the chain rule to a changing inner quantity.
- Interpret an integral as signed accumulation and include its initial condition.
- Separate distance from displacement and work from impulse.
- Distinguish exact model calculations from estimates based on sampled data.
Equations in this section (LaTeX):
W=\int_{x_1}^{x_2}F_x(x)\,dx,\qquad J_x=\int_{t_1}^{t_2}F_x(t)\,dt
Meaning: Work has units N·m = J; impulse has units N·s = kg·m/s. The integration variable makes the difference.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.08 / Rates and accumulation: calculus for physics
Calculus models and real sampled data are not identical
A mathematical function can be differentiated exactly within its model. Experimental data arrive at a finite set of times and carry uncertainty. A finite-difference estimate uses nearby observations; making the time spacing extremely small can amplify fixed position-reading noise when the difference is divided by that small interval.
Numerical integration can use rectangles or trapezia between sampled values. The trapezium rule is exact for a straight line between samples, but not for every curved underlying response. Finer sampling can reduce a discretisation error when the function is sufficiently regular; it does not remove calibration bias or compensate for a rapid event that was never sampled.
At a model boundary, work piecewise. A force may switch off, a particle may collide, or a resistor may change its operating regime. Integrate each valid interval and carry the endpoint condition into the next. Differentiating through an ideal sudden jump may require a more advanced impulsive description; ordinary smooth-function rules cannot simply erase the discontinuity.
WORKED EXAMPLE
What a two-point trapezium assumes
A current is known from a model to be I(t) = t² amperes when t is in seconds. Compare the exact charge transferred from 0 to 2 s with a trapezium estimate using only endpoint currents.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Rates and accumulation: calculus for physics Focus: Calculus models and real sampled data are not identical Learning goals: - Interpret a derivative as a local rate and a tangent slope. - Differentiate simple models while retaining their physical units. - Apply the chain rule to a changing inner quantity. - Interpret an integral as signed accumulation and include its initial condition. - Separate distance from displacement and work from impulse. - Distinguish exact model calculations from estimates based on sampled data. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- A derivative is a limiting rate; it is not obtained by setting both numerator and denominator to zero.
- The derivative’s units are the units of the dependent quantity divided by those of the independent variable.
- The chain rule includes the rate of the inner quantity.
- A definite integral of a rate gives a change; an initial condition gives the absolute state.
- Signed accumulation can cancel. Distance requires the integral of speed.
- Check an antiderivative by differentiating it, then check its initial condition.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
For x(t) = 2t² + 5 metres with t in seconds, what is the velocity at 3 s?
Answer in m/s.
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SELF CHECK 2
A current follows I(t) = 2 + 3t amperes, with t in seconds. What charge passes from 0 to 2 s?
Answer in C.
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SELF CHECK 3
Two particles have the same velocity function but different initial positions. What follows?
- Their positions must become identical
- Their separation stays constant while those velocities remain identical
- One must have a different acceleration
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SELF CHECK 4
Why can an extremely short finite-difference interval be problematic with noisy position data?
- A derivative can never be approximated
- A fixed reading error is divided by a very small time difference
- Acceleration must become infinite
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
A zero derivative hides a finite rangeAdvanced · Hints and guided reasoningZero velocity with a nonzero slope of velocityIntermediate · Hints and guided reasoningAcceleration from a speed-position lawAdvanced · Hints and guided reasoningA growing accelerationIntermediate · Hints and guided reasoningOne graph, two travel measuresIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
A velocity graph has equal areas above and below the time axis. What can you conclude about displacement and distance?
Compare your reasoning
Signed areas cancel, so displacement is zero. Distance adds the magnitudes of the areas and is positive if there was motion. Integration does not discard the sign convention.
Revisit the explanation →A few days later: change the situation
If spring energy is kx²/2 and x changes with time, what extra information is needed to find its rate of change?
Compare your reasoning
The velocity dx/dt is needed. The chain rule gives dU/dt = kx(dx/dt). A steep energy-versus-position curve does not by itself give a time rate.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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