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

The integral from 0 to a of x(a-x) is (a^3)/2-(a^3)/3

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

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

\frac{a ^{3}}{2} - \frac{a ^{3}}{3}

Solution steps

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

Expand

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

= \int_{0}^{a} a 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}^{a} a x d x - \int_{0}^{a} x^{2} d x

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

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

= \frac{a ^{3}}{2} - \frac{a ^{3}}{3}

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What is the integral from 0 to a of x(a-x) ?
The integral from 0 to a of x(a-x) is (a^3)/2-(a^3)/3