220 CHAPTER 9. A FEW STANDARD APPLICATIONS

9.2 Exercises1. Find the length of the graph of y = ln(cosx) for x ∈ [0,π/4] .

2. The curve defined by y = ln(cosx) for x ∈ [0,1] is revolved about the y axis. Find anintegral for the area of the surface of revolution.

3. Find the length of the graph of y = x1/2 − x3/2

3 for x ∈ [1,3] .

4. The graph of the function y = x3 is revolved about the x axis for x ∈ [0,1] . Find thearea of the resulting surface of revolution.

5. The graph of the function y = x3 is revolved about the y axis for x ∈ [0,1] . Find thearea of the resulting surface of revolution. Hint: Formulate this in terms of x and usea change of variables.

6. The graph of the function y = lnx is revolved about the y axis for x ∈ [1,2] . Find thearea of the resulting surface of revolution. Hint: Consider x as a function of y.

7. The graph of the function y = lnx is revolved about the x axis for x ∈ [1,2] . Find thearea of the resulting surface of revolution. If you cannot do the integral, set it up.

8. Find the length of y = cosh(x) for x ∈ [0,1] .

9. Find the length of y = 2x2 − 116 lnx for x ∈ [1,2] .

10. The curve defined by y = 2x2 − 116 lnx for x ∈ [1,2] is revolved about the y axis. Find

the area of the resulting surface of revolution.

11. Find the length of y = x2 − 18 lnx for x ∈ [1,2] .

12. The curve defined by y = x2 − 18 lnx for x ∈ [1,2] is revolved about the y axis. Find

the area of the resulting surface of revolution.

13. The curve defined by y = cosh(x) for x ∈ [0,1] is revolved about the x axis. Find thearea of the resulting surface of revolution.

14. The curve defined by y = cosh(x) for x ∈ [0,1] is revolved about the line y = −3.Find the area of the resulting surface of revolution.

15. For a a positive real number, find the length of y = ax2

2 − 14a lnx for x ∈ [1,2] . Of

course your answer should depend on a.

16. The graph of the function y = x2 for x ∈ [0,1] is revolved about the x axis. Find thearea of the surface of revolution.

17. The graph of the function y =√

x for x ∈ [0,1] is revolved about the y axis. Find thearea of the surface of revolution. Hint: Switch x and y and then use the previousproblem.

18. The graph of the function y = x1/2 − x3/2

3 is revolved about the y axis. Find the areaof the surface of revolution if x ∈ [0,2] .