What is the integral from 4 to 14 of (4-x)(x-14) ?

The integral from 4 to 14 of (4-x)(x-14) is 500/3

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integral from 4 to 14 of (4-x)(x-14)

\int_{4}^{14} (4 - x) (x - 14) d x

\frac{500}{3}

Solution steps

\int_{4}^{14} (4 - x) (x - 14) d x

Expand

(4 - x) (x - 14) : 18 x - x^{2} - 56

= \int_{4}^{14} 18 x - x^{2} - 56 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}^{14} 18 x d x - \int_{4}^{14} x^{2} d x - \int_{4}^{14} 56 d x

\int_{4}^{14} 18 x d x = 1620

\int_{4}^{14} x^{2} d x = \frac{2680}{3}

\int_{4}^{14} 56 d x = 560

= 1620 - \frac{2680}{3} - 560

Simplify

1620 - \frac{2680}{3} - 560 : \frac{500}{3}

= \frac{500}{3}

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What is the integral from 4 to 14 of (4-x)(x-14) ?
The integral from 4 to 14 of (4-x)(x-14) is 500/3