A 5 lb monkey is attached to the end of a 30ft hanging chain that weighs 0.2lb/ft. It climbs the chain to the top. How much work does it do?
There are a few ways to do this problem here is one.
When the monkey is at the bottom the total weight it is carrying is 5 lbs
$\displaystyle (0,5)$ and when it is at the top it is carrying
$\displaystyle 5+15(.2)=8$ beucase it is only holding up half of the chain.
$\displaystyle (30,8)$
The equation between these two points you get
$\displaystyle F(x)=\frac{1}{10}x+5$
So the work is
$\displaystyle \int_{0}^{30}\left( \frac{1}{10}x+5 \right)dx=195 ft \cdot lbs$
I hope this helps. Good luck