What is the integral from-5 to 0 of xsqrt(4-x) ?

The integral from-5 to 0 of xsqrt(4-x) is -506/15

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integral from-5 to 0 of xsqrt(4-x)

\int_{- 5}^{0} x 4 - x d x

- \frac{506}{15}

Decimal

- 33.73333 \dots

Solution steps

\int_{- 5}^{0} x 4 - x d x

Apply u-substitution

= \int_{9}^{4} - (- u + 4) u d u

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

= - \int_{4}^{9} - (- u + 4) u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \int_{4}^{9} (- u + 4) u d u)

Expand

(- u + 4) u : - u^{\frac{3}{2}} + 4 u

= - (- \int_{4}^{9} - u^{\frac{3}{2}} + 4 u d u)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- (- \int_{4}^{9} u^{\frac{3}{2}} d u + \int_{4}^{9} 4 u d u))

\int_{4}^{9} u^{\frac{3}{2}} d u = \frac{422}{5}

\int_{4}^{9} 4 u d u = \frac{152}{3}

= - (- (- \frac{422}{5} + \frac{152}{3}))

Simplify

- (- (- \frac{422}{5} + \frac{152}{3})) : - \frac{506}{15}

= - \frac{506}{15}

What is the integral from-5 to 0 of xsqrt(4-x) ?
The integral from-5 to 0 of xsqrt(4-x) is -506/15