Question

How do you evaluate `\int _ { 2} ^ { 4} 2\pi ( 5- x ) x ^ { 2} d x`?

Answer 1

`200/3 pi`

Explanation:

`\int _ { 2} ^ { 4} 2\pi ( 5- x ) x ^ { 2} d x = 2 pi\int _ { 2} ^ { 4} ( 5x^2- x^3 ) d x`
`qquad = 2pi (5/3x^3-1/4x^4)_2^4=2pi [5/3(4^3-2^3)-1/4(4^2-2^2)]`
`qquad = 2 pi 100/3=200/3 pi`