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 6.1.5-6.
  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 us a computer algebra system (CAS) like Derive, or the TI89 or TI92 to evaluate the integrals. Look at problems 6.1.7-14, 6.2.5-20, 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 6.4.3-8 and 6.5.1-8.
  4. Work problem involving springs. Look at problems 6.6.4-7.
  5. Work problem. Look at problems 6.6.8-17.
  6. Find the force and pressure on a horizontally submerged surface. Find the fluid force on a vertically submerged surface. Look at problems 6.7.1-8.
  7. A region is given. Find the volume of rotation when the region is rotated about a line other than a coordinate axes. Write a definite integral and then use a computer algebra system to evaluate the integral. Look at problems 6.2.25-28 and 6.3.18-19.
  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 6.2.39.

Notes

Point values per problem

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