What is the integral from 0 to 8 of xsqrt(8-x) ?

The integral from 0 to 8 of xsqrt(8-x) is (512sqrt(2))/(15)

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integral from 0 to 8 of xsqrt(8-x)

\int_{0}^{8} x 8 - x d x

\frac{512 2}{15}

Decimal Notation

48.27182 \dots

Solution steps

\int_{0}^{8} x 8 - x d x

Apply u-substitution

: \int_{8}^{0} - (- u + 8) u d u

= \int_{8}^{0} - (- u + 8) u d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{8} - (- u + 8) u d u

Take the constant out:

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

= - (- \int_{0}^{8} (- u + 8) u d u)

Expand

(- u + 8) u : - u^{\frac{3}{2}} + 8 u

= - (- \int_{0}^{8} - u^{\frac{3}{2}} + 8 u d u)

Apply the Sum Rule:

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

= - (- (- \int_{0}^{8} u^{\frac{3}{2}} d u + \int_{0}^{8} 8 u d u))

\int_{0}^{8} u^{\frac{3}{2}} d u = \frac{256 2}{5}

\int_{0}^{8} 8 u d u = \frac{256 2}{3}

= - (- (- \frac{256 2}{5} + \frac{256 2}{3}))

Simplify

- (- (- \frac{256 2}{5} + \frac{256 2}{3})) : \frac{512 2}{15}

= \frac{512 2}{15}

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What is the integral from 0 to 8 of xsqrt(8-x) ?
The integral from 0 to 8 of xsqrt(8-x) is (512sqrt(2))/(15)