What is the integral from 0 to a of sqrt(a^2-x^2)x ?

The integral from 0 to a of sqrt(a^2-x^2)x is ((a^2)^{3/2})/3

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integral from 0 to a of sqrt(a^2-x^2)x

\int_{0}^{a} a^{2} - x^{2} x d x

\frac{( a ^{2} ) ^{\frac{3}{2}}}{3}

Solution steps

\int_{0}^{a} a^{2} - x^{2} x d x

Apply u-substitution

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

= \int_{a^{2}}^{0} - \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_{a^{2}}^{0} u d u

Apply the Power Rule

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

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

Compute the boundaries:

- \frac{2}{3} (a^{2})^{\frac{3}{2}}

= - \frac{1}{2} (- \frac{2}{3} (a^{2})^{\frac{3}{2}})

Simplify

= \frac{( a ^{2} ) ^{\frac{3}{2}}}{3}

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What is the integral from 0 to a of sqrt(a^2-x^2)x ?
The integral from 0 to a of sqrt(a^2-x^2)x is ((a^2)^{3/2})/3