代写math3B Worksheet week 5帮做R程序

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