# Thread: Applications of Integrals

1. ## Applications of Integrals

I'm having trouble starting this problem out. I don't know where to begin. Could anyone point me in the right direction or link me to a tutorial?

Find the work done in pumping all the water out of a conical
reservoir of radius of base 3 ft and height 5 ft, if the water is
to be lifted one foot above the top of the reservoir.

2. Originally Posted by maxreality
I'm having trouble starting this problem out. I don't know where to begin. Could anyone point me in the right direction or link me to a tutorial?

Find the work done in pumping all the water out of a conical
reservoir of radius of base 3 ft and height 5 ft, if the water is
to be lifted one foot above the top of the reservoir.
Start by drawing the picture.

3. You picture (of the reservoir seen from the side) should look like two intersecting diagonal lines with a horizontal line connecting them at the bottom. Since the sides are lines, the distance "across" at any point is proportional to the distance "down" ("similar triangles"). You can use the fact that the distance across at the top is 3 and the distance across is 3 at "down 5" to determine the proportionality.

Now imagine a "layer of water" at depth z. It is a disk with radius half the distance across and depth "dz". Its volume is $\displaystyle \pi r^2 dz$ and its weight is that times its density. To find the work done is lifting that layer, multiply its weight by the distance lifted- z+ 1 since you are to lift it to a point "one foot above the top of the reservoir".

To find the work done to lift all of the water, integrate over the depth of the rezervoir.