What is the integral from 0 to 2 of 2pi(x-3)x^2 ?

The integral from 0 to 2 of 2pi(x-3)x^2 is -8pi

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integral from 0 to 2 of 2pi(x-3)x^2

\int_{0}^{2} 2 π (x - 3) x^{2} d x

- 8 π

Solution steps

\int_{0}^{2} 2 π (x - 3) x^{2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 π \cdot \int_{0}^{2} (x - 3) x^{2} d x

Expand

(x - 3) x^{2} : x^{3} - 3 x^{2}

= 2 π \cdot \int_{0}^{2} x^{3} - 3 x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 π (\int_{0}^{2} x^{3} d x - \int_{0}^{2} 3 x^{2} d x)

\int_{0}^{2} x^{3} d x = 4

\int_{0}^{2} 3 x^{2} d x = 8

= 2 π (4 - 8)

Simplify

= - 8 π

\int_{- 1}^{1} (x^{3} - 3 x^{2} - x + 3) d x

\int_{0}^{1} \frac{2}{x ^{2}} d x

\int_{2}^{6} - x^{2} + 8 d x

\int_{0}^{2 π} 1 + sin (θ) d θ

\int_{- 2}^{0} 5 x^{4} - 3 x^{2} + x - 3 d x

What is the integral from 0 to 2 of 2pi(x-3)x^2 ?
The integral from 0 to 2 of 2pi(x-3)x^2 is -8pi