What is the integral from 2 to 4 of (x-2)(4-x) ?

The integral from 2 to 4 of (x-2)(4-x) is 4/3

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integral from 2 to 4 of (x-2)(4-x)

\int_{2}^{4} (x - 2) (4 - x) d x

\frac{4}{3}

Decimal Notation

1.33333 \dots

Solution steps

\int_{2}^{4} (x - 2) (4 - x) d x

Expand

(x - 2) (4 - x) : 6 x - x^{2} - 8

= \int_{2}^{4} 6 x - x^{2} - 8 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}^{4} 6 x d x - \int_{2}^{4} x^{2} d x - \int_{2}^{4} 8 d x

\int_{2}^{4} 6 x d x = 36

\int_{2}^{4} x^{2} d x = \frac{56}{3}

\int_{2}^{4} 8 d x = 16

= 36 - \frac{56}{3} - 16

Simplify

36 - \frac{56}{3} - 16 : \frac{4}{3}

= \frac{4}{3}

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What is the integral from 2 to 4 of (x-2)(4-x) ?
The integral from 2 to 4 of (x-2)(4-x) is 4/3