Math 121 - Exam 6 Study Guide

1. Two curves are given along with a graph of the region enclosed between the curves. Write an integral with respect to x that can be used to find the area between the curves and write an integral with respect to y that can be used to find the area between the curves. Note that depending on the region, you may need more than one integral to accomplish this. Evaluate whichever integral is easier to find the area between the curves. Look at problems 6.1.1-6.
2. Sketch the region enclosed by two curves. Write definite integrals that can be used to find the volume of rotation about each of the coordinate axes and then use a computer algebra system (CAS) like Derive, or the TI89 or TI92 to evaluate the integrals. Look at problems 6.2.7-11, 15-20 and 6.3.5-14.
3. A parametically defined curve is given along with a sketch of the curve. Find and simplify the arc length parameter sqrt [ (dx/dt)2 + (dy/dt)2 ]. Then use the results to find the length of the curve and the surface area of the surface generated when the curve is rotated about each of the coordinate axes. In each case, setup a definite integral and then use a computer algebra system (TI89, TI92, or Derive) to evaluate or approximate the integrals. Look at problems 6.4.9-12 and 6.5.27-30.
4. Work problem involving springs. Look at problems 6.7.6-11.
5. Work problem. Look at problems 6.7.12-21.
6. Find the force and pressure on a horizontally submerged surface. Find the fluid force on a vertically submerged surface. Look at problems 6.8.1-8.
7. Find the average value of a function over an interval. Look at problems 6.6.3-8.
8. The graph of an unknown function is shown along with key points on the curve. Use trapezoids to approximate the area under the curve. Make a table of the midpoints between the left and right endpoints of each interval, the change in x and y and the length of the secant line segment for each interval. Use the table to find the volume of the solid generated when the region is rotated about the x-axis and the y-axis. Approximate the length of the curve and the surface area of revolution when the curve is rotated about one of the coordinate axes. Look at problem 6.2.37.

Notes

• Show all work and give exact answers except where indicated.
• You should know the weight density of fresh water in the English and metric systems.
• There are some problems (indicated on the test) where you are to set up the integral and then use the TI89, TI92, or Derive to find the exact answers. You do not need to show work on those problems after you have the definite integral set up. But make sure that you enter the integral correctly into the algebra system.
• There aren't many questions, so each one is worth a lot of points. In particular, pay attention to #8. It's actually one of the easier problems on the test if you understand the concepts but it is designed to make sure you understand the concepts and haven't just memorized a bunch of formulas.

Point values per problem

 # Pts 1 2 3 4 5 6 7 8 Total 10 12 13 8 8 10 6 33 100