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

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

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integral from 0 to h of (L^2)/(h^2)x^2

\int_{0}^{h} \frac{L ^{2}}{h ^{2}} x^{2} d x

\frac{h L ^{2}}{3}

Solution steps

\int_{0}^{h} \frac{L ^{2}}{h ^{2}} x^{2} d x

Take the constant out:

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

= \frac{L ^{2}}{h ^{2}} \cdot \int_{0}^{h} x^{2} d x

Apply the Power Rule

: [\frac{x ^{3}}{3}]_{0}^{h}

= \frac{L ^{2}}{h ^{2}} [\frac{x ^{3}}{3}]_{0}^{h}

Compute the boundaries:

\frac{h ^{3}}{3}

= \frac{L ^{2}}{h ^{2}} \cdot \frac{h ^{3}}{3}

Simplify

= \frac{h L ^{2}}{3}

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