What is the integral from 0 to 3 of 2pi(10-x)(2x) ?

The integral from 0 to 3 of 2pi(10-x)(2x) is 144pi

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integral from 0 to 3 of 2pi(10-x)(2x)

\int_{0}^{3} 2 π (10 - x) (2 x) d x

144 π

Decimal

452.38934 \dots

Solution steps

\int_{0}^{3} 2 π (10 - x) \cdot 2 x d x

Take the constant out:

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

= 2 π 2 \cdot \int_{0}^{3} (10 - x) x d x

Simplify

= 4 π \cdot \int_{0}^{3} (10 - x) x d x

Expand

(10 - x) x : 10 x - x^{2}

= 4 π \cdot \int_{0}^{3} 10 x - 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

= 4 π (\int_{0}^{3} 10 x d x - \int_{0}^{3} x^{2} d x)

\int_{0}^{3} 10 x d x = 45

\int_{0}^{3} x^{2} d x = 9

= 4 π (45 - 9)

Simplify

= 144 π

What is the integral from 0 to 3 of 2pi(10-x)(2x) ?
The integral from 0 to 3 of 2pi(10-x)(2x) is 144pi