If the curve is rotated about a horizontal line ...

Question

If the curve y = f(x), in which x is between a and b, is rotated around the horizontal line y=c, then the formula used for calculating the surface area is:

S = ∫ (a to b) 2π[c-f(x)]√(1+[f ' (x)]²)dx

I understand this perfectly fine, I've done plenty of problems with it. The issue is finding the area of the surface generated by rotating the circle x² + y² = r² about the line y = r. Substituting values gives an incredibly long integral that I don't know how to solve.

Answer 1

S = 2π∫_-r^r(r- √r² - x²)√[1 + x²/(r²- x²)]dx + 2π∫_-r^r(r + √r² - x²)√[1 + x²/(r²- x²)]dx =

4πr∫_-r^r √r²/(r² - x²)dx = 4πr²⌈_-r^r(r²- x²)^-1/2dx= 8πr²∫₀^r(r² - x²)^-1/2dx; x = rsinθ; dx = rcosθdθ; x = 0 θ = 0;

S = 8πr²∫₀^π/2dθ = 4π²r².