What is the integral from 2 to 3 of sqrt(9-r^2)r ?

The integral from 2 to 3 of sqrt(9-r^2)r is (5sqrt(5))/3

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integral from 2 to 3 of sqrt(9-r^2)r

\int_{2}^{3} 9 - r^{2} r d r

\frac{5 5}{3}

Decimal

3.72677 \dots

Solution steps

\int_{2}^{3} 9 - r^{2} r d r

Apply u-substitution

: \int_{5}^{0} - \frac{u}{2} d u

= \int_{5}^{0} - \frac{u}{2} d u

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

= - \int_{0}^{5} - \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}^{5} u d u)

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{5}

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{5})

Simplify

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{5}

Compute the boundaries:

\frac{10 5}{3}

= \frac{1}{2} \cdot \frac{10 5}{3}

Simplify

= \frac{5 5}{3}

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What is the integral from 2 to 3 of sqrt(9-r^2)r ?
The integral from 2 to 3 of sqrt(9-r^2)r is (5sqrt(5))/3