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

The integral from 0 to 2pi of (θ^2)/2 is (4pi^3)/3

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integral from 0 to 2pi of (θ^2)/2

\int_{0}^{2 π} \frac{θ ^{2}}{2} d θ

\frac{4 π ^{3}}{3}

Decimal

41.34170 \dots

Solution steps

\int_{0}^{2 π} \frac{θ ^{2}}{2} d θ

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{2 π} θ^{2} d θ

Apply the Power Rule

: [\frac{θ ^{3}}{3}]_{0}^{2 π}

= \frac{1}{2} [\frac{θ ^{3}}{3}]_{0}^{2 π}

Compute the boundaries:

\frac{8 π ^{3}}{3}

= \frac{1}{2} \cdot \frac{8 π ^{3}}{3}

Simplify

= \frac{4 π ^{3}}{3}

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What is the integral from 0 to 2pi of (θ^2)/2 ?
The integral from 0 to 2pi of (θ^2)/2 is (4pi^3)/3