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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

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

Solution steps

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

Expand

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

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

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

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

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

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