Work of a variable force

Text

An object is moved along a line from $x=0$ to $x=4\text{ m}$ under a force $F(x)=3x^{2}+2$ (newtons, directed along the motion). Compute the work done.

Solution

$W=\displaystyle\int_0^4 (3x^{2}+2)\,dx=\big[x^{3}+2x\big]_0^4=72\text{ J}$. $\boxed{W=72\ \text{J}}$