05 - Pumping water from a tank
A cylindrical tank is full of water and the water is pumped out the top. The length of the tank is 7m and the radius is 5m.
- (a) Set up an integral for the total work performed assuming the tank is initially completely full.
- (b) Set up an integral for the total work performed assuming the tank is initially full to 3m and is pumped up to a height of 1m before exiting the tank.
Solution
(a)
Identify line
Set
Make the top of the cylinder
Find formula for weight of a single layer.
Area of a layer at
One layer is a rectangle with length 7m.
The width is directly related to the formula for a circle of radius 5.
Volume of a layer at
Weight of the layer is then
Find formula for vertical distance a plate is lifted.
Layer at
Set up integral.
(b) Find formula for vertical distance a plate is lifted.
Layer at
Adjust bounds.
Since the tank is full 3m deep, the water starts at
Set up integral.
06 - Work required to build a pyramid
The Great Pyramid of Giza is 140m tall and has a square base with 230m on each side. It is built of stone with mass density 2000kg/m
Solution
Identify line
Set
Let the top of the pyramid be at
Find formula for weight of a single layer.
At
At
The formula for the side is thus
The area is
Weight would be
Set up integral.
09 - Computing improper integrals, Part I
For each integral below, give the limit interpretation and compute that limit. Based on that result, state whether the integral converges. If it converges, what is its value?
- (a)
. - (b)
. - (c)
Solution (a)
Rewrite using limits.
Compute integral.
(b)
Rewrite using limits. (
Compute integral.
(c)
Rewrite using limits.
Compute integral.
10 - Computing improper integrals, Part II
- (a)
- (b)
- (c)
Solution (a)
Rewrite using limits.
Compute integral.
(b) Rewrite using limits.
Compute integrals.
(c)
Rewrite using limits.
Compute integral.