How to evaluate a definite integral..

Question

How would you write and evaluate a definite integral that represents volume of a solid formed by revolving it's region about the x-axis for y = −x+8?

Answer 1

pi * integral ( -x+8)^2 =

pi * integral (x^2 -16x + 64) =

pi * [ 1/3 * x^3 - 8x^2 + 64x ]

Now you need the limits of integration.

The x-intercept is (8,0), so x=[0,8] sounds good

pi * [ 1/3 * 512 - 512 + 512]

pi * 512/3 =

(pi/3)*512