What is the integral from 0 to 20 of x(20-x) ?

The integral from 0 to 20 of x(20-x) is 4000/3

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integral from 0 to 20 of x(20-x)

\int_{0}^{20} x (20 - x) d x

\frac{4000}{3}

Decimal Notation

1333.33333 \dots

Solution steps

\int_{0}^{20} x (20 - x) d x

Expand

x (20 - x) : 20 x - x^{2}

= \int_{0}^{20} 20 x - x^{2} 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_{0}^{20} 20 x d x - \int_{0}^{20} x^{2} d x

\int_{0}^{20} 20 x d x = 4000

\int_{0}^{20} x^{2} d x = \frac{8000}{3}

= 4000 - \frac{8000}{3}

Simplify

4000 - \frac{8000}{3} : \frac{4000}{3}

= \frac{4000}{3}

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What is the integral from 0 to 20 of x(20-x) ?
The integral from 0 to 20 of x(20-x) is 4000/3