What is the integral from 8 to 12 of (8-x)(x-12) ?

The integral from 8 to 12 of (8-x)(x-12) is 32/3

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integral from 8 to 12 of (8-x)(x-12)

\int_{8}^{12} (8 - x) (x - 12) d x

\frac{32}{3}

Decimal

10.66666 \dots

Solution steps

\int_{8}^{12} (8 - x) (x - 12) d x

Expand

(8 - x) (x - 12) : 20 x - x^{2} - 96

= \int_{8}^{12} 20 x - x^{2} - 96 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_{8}^{12} 20 x d x - \int_{8}^{12} x^{2} d x - \int_{8}^{12} 96 d x

\int_{8}^{12} 20 x d x = 800

\int_{8}^{12} x^{2} d x = \frac{1216}{3}

\int_{8}^{12} 96 d x = 384

= 800 - \frac{1216}{3} - 384

Simplify

800 - \frac{1216}{3} - 384 : \frac{32}{3}

= \frac{32}{3}

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What is the integral from 8 to 12 of (8-x)(x-12) ?
The integral from 8 to 12 of (8-x)(x-12) is 32/3