Worksheet week 5
Set up (do not evaluate) the volume integral.
1. Rotate the region bounded by y = ln x, y = 0, x = 3 about the x-axis.
Set up a single integral using washers/disks: π( )2 dx.
2. Rotate the region bounded by about the y-axis.
Set up a single integral using washers/disks (in y): ——π( )2 dx.
Let S be the solid obtained by rotating the shaded region under y = sin(x2) from x = 0 to about the y-axis.
a) Sketch a typical cylindrical shell at position x. What are its circumference and height?
Circumference: Height:
b) Use shells to set up (and then evaluate) the volume V.
dx
c) Is the shell method preferable to washers here? Briefly explain.
Integration by Parts & Friends
3.
4.
5.
6. Hint: Use a u-sub first.
7. (Reduction) Use integration by parts to prove
8. Use the reduction result with n = 3 to find