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

The integral from 4 to 5 of (x-4)(5-x) is 1/6

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

\int_{4}^{5} (x - 4) (5 - x) d x

\frac{1}{6}

Decimal Notation

0.16666 \dots

Solution steps

\int_{4}^{5} (x - 4) (5 - x) d x

Expand

(x - 4) (5 - x) : 9 x - x^{2} - 20

= \int_{4}^{5} 9 x - x^{2} - 20 d x

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}^{5} 9 x d x - \int_{4}^{5} x^{2} d x - \int_{4}^{5} 20 d x

\int_{4}^{5} 9 x d x = \frac{81}{2}

\int_{4}^{5} x^{2} d x = \frac{61}{3}

\int_{4}^{5} 20 d x = 20

= \frac{81}{2} - \frac{61}{3} - 20

Simplify

\frac{81}{2} - \frac{61}{3} - 20 : \frac{1}{6}

= \frac{1}{6}

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What is the integral from 4 to 5 of (x-4)(5-x) ?
The integral from 4 to 5 of (x-4)(5-x) is 1/6