Pappus's Theorem and Some Integrals

Hey guys, I was wondering if it would be possible for someone to check my work on a few calculus problems and, if they're wrong, to tell me what I'm doing wrong. Thank you very much in advance.

**Problem 1.** R is the region bounded by y = sinx and the lines x = 0 and x = .

**a.** Calculate the center of mass of the region R.

**My Solution**

dx

u = x, du = dx, dv = sinx dx, v = cosx

-xcosx + cosx dx = -xcosx + sinx

= = 1

dx = dx = dx = = = =

**b.** Use Pappus's Theorem to calculate the volume of the solid of revolution if R is revolved about the line x = 5.

**My Solution**

Area of region R = 1 (from problem 1a)

V = = =

**c.** Calculate the volume of the solid if R is revolved about the line y = -4.

**My Solution**

Area of region R = 1 (from problem 1a)

V = = = ** **

Problem 2. Calculate each antiderivative.

**a.** dx

My Solution

x = , dx = d , =

= d ** = ** d = d = =

** b.** dx

**My Solution**

u = 1 - , du = -2x dx, dx = du

= du = du = du = =

**c.** dx

**My Solution**

, ,

= = d = d = = = = =

**d.** dx

My Solution

, ,

= = = = = = = = =

Once again, thank you very much in advance.