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

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

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integral from 0 to θ of (2x^3)/(θ^2)

\int_{0}^{θ} \frac{2 x ^{3}}{θ ^{2}} d x

\frac{θ ^{2}}{2}

Solution steps

\int_{0}^{θ} \frac{2 x ^{3}}{θ ^{2}} d x

Take the constant out:

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

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

Apply the Power Rule

: [\frac{x ^{4}}{4}]_{0}^{θ}

= \frac{2}{θ ^{2}} [\frac{x ^{4}}{4}]_{0}^{θ}

Compute the boundaries:

\frac{θ ^{4}}{4}

= \frac{2}{θ ^{2}} \cdot \frac{θ ^{4}}{4}

Simplify

= \frac{θ ^{2}}{2}

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