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

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

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

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

- 243

Solution steps

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

Apply u-substitution

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

= \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}

Compute the boundaries:

486

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

Simplify

= - 243

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