What is the integral from 0 to 4 of xsqrt(23-x^2) ?

The integral from 0 to 4 of xsqrt(23-x^2) is (23sqrt(23)-7sqrt(7))/3

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integral from 0 to 4 of xsqrt(23-x^2)

\int_{0}^{4} x 23 - x^{2} d x

\frac{23 23 - 7 7}{3}

Decimal

30.59462 \dots

Solution steps

\int_{0}^{4} x 23 - x^{2} d x

Apply u-substitution

= \int_{23}^{7} - \frac{u}{2} d u

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

= - \int_{7}^{23} - \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_{7}^{23} u d u)

Apply the Power Rule

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{7}^{23})

Simplify

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{7}^{23}

Compute the boundaries:

\frac{46 23 - 14 7}{3}

= \frac{1}{2} \cdot \frac{46 23 - 14 7}{3}

Simplify

= \frac{23 23 - 7 7}{3}

What is the integral from 0 to 4 of xsqrt(23-x^2) ?
The integral from 0 to 4 of xsqrt(23-x^2) is (23sqrt(23)-7sqrt(7))/3