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

The integral from 0 to 3 of 2pi(x)(3x-x^2) is (27pi)/2

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

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

\frac{27 π}{2}

Decimal

42.41150 \dots

Solution steps

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

Expand

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

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

\int_{0}^{3} 3 x^{2} d x = 27

\int_{0}^{3} x^{3} d x = \frac{81}{4}

= 2 π (27 - \frac{81}{4})

Simplify

2 π (27 - \frac{81}{4}) : \frac{27 π}{2}

= \frac{27 π}{2}

What is the integral from 0 to 3 of 2pi(x)(3x-x^2) ?
The integral from 0 to 3 of 2pi(x)(3x-x^2) is (27pi)/2