Page 63 - Calculus for the Clueless, Calc II

If the can were three-fourths full, integral limits would be 0 to 15.

Note 3

If we did the same problem with a box, the cross section would be a thin sheet with length and width constant
and height dy.

Example 3—

How much work is done to pump water out of a cone, diameter 22 feet, height 10 feet, if the outlet is 7 feet over
the top of the cone and the cone is filled 2 feet deep?

First, a slight trick. The diameter is 22, so the radius is 11. Next, note the cross section is again a thin cylinder,
2
2
but this time the radius changes: V = πr h = πx dy. We must see a similar triangle x/y = 11/10 or x = 11y/10.
The height of the pipe makes the pumping distance 17 - y.
We leave the integration to you.

Note

A trough is a similar problem except the horizontal cross section is a rectangular sheet whose width changes but
whose length (length of the trough) remains the same.

Also note similar triangles, just like in the cone.