Spring 2005 MATH 172
Week in Review III
courtesy of David J. Manuel
Section 7.3-7.5
Section 7.3
1. Write a Riemann Sum definition to find the volume of the solid obtained by rotating the region
bounded by x = 2y − y 2 and x = 0 about the x-axis.
2. Given f (x) ≥ 0 on [a, b], write an integral formula to find the volume of the solid formed by
rotating the region bounded by y = f (x), y = 0, x = a, and x = b about the line x = k, where k < a.
3. A donut is formed by rotating the circle (x − 2)2 + y 2 = 1 about the y-axis. Find the volume.
Section 7.4
4. If a force acts on an object with magnitude F (x) when the object is at position x along a straight
line, state and derive a formula for finding the work done by this force in moving an object from
x = a to x = b.
5. Write a Riemann Sum to find the work required to pump all the water out of a tank 10 meters
long that has ends the shape of a semicircle as shown below, if the tank is full.
10 m
Section 7.5
6. Derive the formula for the average value of f on [a, b].
7. The graph of a function f is shown below. Determine whether the average value of f is less than,
greater than, or equal to 3/2 if possible. Explain why.
2
1.8
1.6
1.4
1.2
y
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
x
8. State the Mean Value Theorem for Integrals. Use the Mean Value Theorem on an appropriate
function to prove the MVT for Integrals.