What is the integral from 4 to 5 of xsqrt(25-x^2) ?

The integral from 4 to 5 of xsqrt(25-x^2) is 9

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integral from 4 to 5 of xsqrt(25-x^2)

\int_{4}^{5} x 25 - x^{2} d x

9

Solution steps

\int_{4}^{5} x 25 - x^{2} d x

Apply u-substitution

= \int_{9}^{0} - \frac{u}{2} d u

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

= - \int_{0}^{9} - \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}^{9} u d u)

Apply the Power Rule

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9})

Simplify

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9}

Compute the boundaries:

18

= \frac{1}{2} \cdot 18

Simplify

= 9

What is the integral from 4 to 5 of xsqrt(25-x^2) ?
The integral from 4 to 5 of xsqrt(25-x^2) is 9