What is the integral from 2 to 6 of (2-x)(x-6) ?

The integral from 2 to 6 of (2-x)(x-6) is 32/3

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integral from 2 to 6 of (2-x)(x-6)

\int_{2}^{6} (2 - x) (x - 6) d x

\frac{32}{3}

Decimal

10.66666 \dots

Solution steps

\int_{2}^{6} (2 - x) (x - 6) d x

Expand

(2 - x) (x - 6) : 8 x - x^{2} - 12

= \int_{2}^{6} 8 x - x^{2} - 12 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_{2}^{6} 8 x d x - \int_{2}^{6} x^{2} d x - \int_{2}^{6} 12 d x

\int_{2}^{6} 8 x d x = 128

\int_{2}^{6} x^{2} d x = \frac{208}{3}

\int_{2}^{6} 12 d x = 48

= 128 - \frac{208}{3} - 48

Simplify

128 - \frac{208}{3} - 48 : \frac{32}{3}

= \frac{32}{3}

What is the integral from 2 to 6 of (2-x)(x-6) ?
The integral from 2 to 6 of (2-x)(x-6) is 32/3