Math 182

Test 2 review sheet

This test will emphasize chapter 6 – applications of integrals. That doesn't mean that material from the first test won't show up (if you forget how to integrate, you can expect to find this quite difficult!) - it just means that I won't be focusing on material from back then.

Remember, the best way to prepare for the test is to re-read your notes and the book, and do several problems from each section. Get stuck? Keep trying until you figure it out, or until your head starts to really hurt from being banged against the wall over and over again! Then come see me.

Want still more problems? Good – here are some. As with the last review, most of these questions are either based on, or even taken straight from, tests I have previously given.

1. Consider the region R bounded by

a. What is the area of R?

b. What is the volume of the solid formed by revolving R about the x-axis?

c. What is the volume of the solid formed by revolving R about the line x= -1?

d. What is the volume of the solid formed by revolving R about the y-axis?

e. What is perimeter of R?

2. The base of a region is the unit circle. Cross sections perpendicular to the base are squares. What is the volume of this region?

3. a. What is the average value of the function for ?

b. The Mean Value Theorem for Integrals guarantees that there is a number c in the domain such that , the average value found in part (a). What is that number c? Geometrically, what is the significance of this number?

4. A tank is in the shape of a cone as shown, with radius at the base 5 m, and height 10 m.

If the tank is initially filled to a height of 8m with water, how much work is required to pump all of the water out of the top? Recall that water has a density of .

5. The force exerted by the earth’s gravity on a 1 kg mass is , where r is the distance of the mass from the center of the earth. (Note: the radius of the earth is approx. ).

a. According to this formula, what is the weight of the 1 kg mass on the surface of the earth?

b. What is the weight of the mass 1000 km above the surface of the earth?

c. How much work is required to launch this mass from the surface of the earth to a height of 1000 km?

6. Consider a plate of uniform density bounded by , the x-axis, and . Find the moments and center of mass of this plate.

7. The demand function for a certain commodity is . If the price is set at $5, determine the quantity that will be sold, the revenue, and the consumer surplus.