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

The integral from 0 to 2 of pi(x^3-8)^2 is pi 576/7

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

\int_{0}^{2} π (x^{3} - 8)^{2} d x

π \frac{576}{7}

Decimal

258.50819 \dots

Solution steps

\int_{0}^{2} π (x^{3} - 8)^{2} d x

Take the constant out:

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

= π \cdot \int_{0}^{2} (x^{3} - 8)^{2} d x

Expand

(x^{3} - 8)^{2} : x^{6} - 16 x^{3} + 64

= π \cdot \int_{0}^{2} x^{6} - 16 x^{3} + 64 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

= π (\int_{0}^{2} x^{6} d x - \int_{0}^{2} 16 x^{3} d x + \int_{0}^{2} 64 d x)

\int_{0}^{2} x^{6} d x = \frac{128}{7}

\int_{0}^{2} 16 x^{3} d x = 64

\int_{0}^{2} 64 d x = 128

= π (\frac{128}{7} - 64 + 128)

Simplify

π (\frac{128}{7} - 64 + 128) : π \frac{576}{7}

= π \frac{576}{7}

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What is the integral from 0 to 2 of pi(x^3-8)^2 ?
The integral from 0 to 2 of pi(x^3-8)^2 is pi 576/7