What is the integral from 0 to L of (L-x)^2 ?

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

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integral from 0 to L of (L-x)^2

\int_{0}^{L} (L - x)^{2} d x

\frac{L ^{3}}{3}

Solution steps

\int_{0}^{L} (L - x)^{2} d x

Apply u-substitution

= \int_{L}^{0} - u^{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int_{L}^{0} u^{2} d u

Apply the Power Rule

= - [\frac{u ^{3}}{3}]_{L}^{0}

Compute the boundaries:

- \frac{L ^{3}}{3}

= - (- \frac{L ^{3}}{3})

Simplify

= \frac{L ^{3}}{3}

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