What is the integral from 0 to 9 of xsqrt(81-x^2) ?

The integral from 0 to 9 of xsqrt(81-x^2) is 243

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integral from 0 to 9 of xsqrt(81-x^2)

\int_{0}^{9} x 81 - x^{2} d x

243

Solution steps

\int_{0}^{9} x 81 - x^{2} d x

Apply u-substitution

: \int_{81}^{0} - \frac{u}{2} d u

= \int_{81}^{0} - \frac{u}{2} d u

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

= - \int_{0}^{81} - \frac{u}{2} d u

Take the constant out:

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

= - (- \frac{1}{2} \cdot \int_{0}^{81} u d u)

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{81}

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{81})

Simplify

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{81}

Compute the boundaries:

486

= \frac{1}{2} \cdot 486

Simplify

= 243

\frac{d}{d x} (e^{2 x} + 2)

\frac{s ^{2}}{s + 1}

x y^{^{'}} = y + 2 x^{2} sin (x), y (π) = 0

n = 1 \sum \infty \frac{e ^{n}}{n ^{2}}

\frac{\partial}{\partial x} (2 x^{3} + 3 y^{3} - 12 x y)

What is the integral from 0 to 9 of xsqrt(81-x^2) ?
The integral from 0 to 9 of xsqrt(81-x^2) is 243