What is the integral from 3 to 7 of (3-x)(x-7) ?

The integral from 3 to 7 of (3-x)(x-7) is 32/3

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integral from 3 to 7 of (3-x)(x-7)

\int_{3}^{7} (3 - x) (x - 7) d x

\frac{32}{3}

Decimal

10.66666 \dots

Solution steps

\int_{3}^{7} (3 - x) (x - 7) d x

Expand

(3 - x) (x - 7) : 10 x - x^{2} - 21

= \int_{3}^{7} 10 x - x^{2} - 21 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_{3}^{7} 10 x d x - \int_{3}^{7} x^{2} d x - \int_{3}^{7} 21 d x

\int_{3}^{7} 10 x d x = 200

\int_{3}^{7} x^{2} d x = \frac{316}{3}

\int_{3}^{7} 21 d x = 84

= 200 - \frac{316}{3} - 84

Simplify

200 - \frac{316}{3} - 84 : \frac{32}{3}

= \frac{32}{3}

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What is the integral from 3 to 7 of (3-x)(x-7) ?
The integral from 3 to 7 of (3-x)(x-7) is 32/3