f(x) = √(r² - x²)
the formula for the surface area rotated about the x-axis is
S = 2π ∫[-r,r] f(x) √(1 + f'(x)²) dx
f '(x) = -x/√(r² - x²)
thus
√(1 + [f'(x)]²) = √(1 + x²/(r²-x²))
= √(r²/(r²-x²))
= r/√(r^2 - x^2)
thus
S = 2π ∫[-r,r] r dx
= 2π (rx) ... from -r to r
= 2π r (2r)
= 4π r²
另外的解法：
1v=∫[0,r] A( r) dr,v是体积
另外：公式