How do you add up a changing force?

A particle moves along the positive x-axis from x = 0 to x = 3 m. A force parallel to the axis has component F(x) = (4 N/m)x.

The force varies with position. The shaded area represents work.

How much work does this force do?

Organize the givens & assumptions

Start: x = 0
End: x = 3 m
Force grows linearly from 0 to 12 N

The expression gives this force’s x-component throughout the displacement; other forces, if present, are irrelevant to the work of this particular force.

A $6\ \mathrm J$
B $12\ \mathrm J$
C $36\ \mathrm J$
D $18\ \mathrm J$

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