What is the integral from 0 to 2 of xsqrt(7-x^2) ?

The integral from 0 to 2 of xsqrt(7-x^2) is (7sqrt(7)-3sqrt(3))/3

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integral from 0 to 2 of xsqrt(7-x^2)

\int_{0}^{2} x 7 - x^{2} d x

\frac{7 7 - 3 3}{3}

Decimal Notation

4.44136 \dots

Solution steps

\int_{0}^{2} x 7 - x^{2} d x

Apply u-substitution

: \int_{7}^{3} - \frac{u}{2} d u

= \int_{7}^{3} - \frac{u}{2} d u

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

= - \int_{3}^{7} - \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_{3}^{7} u d u)

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{3}^{7}

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{3}^{7})

Simplify

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{3}^{7}

Compute the boundaries:

\frac{14 7}{3} - 23

= \frac{1}{2} (\frac{14 7}{3} - 23)

Simplify

= \frac{7 7 - 3 3}{3}

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What is the integral from 0 to 2 of xsqrt(7-x^2) ?
The integral from 0 to 2 of xsqrt(7-x^2) is (7sqrt(7)-3sqrt(3))/3