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

The integral from 0 to 2pi of ((2θ^3)/3) is (8pi^4)/3

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integral from 0 to 2pi of ((2θ^3)/3)

\int_{0}^{2 π} (\frac{2 θ ^{3}}{3}) d θ

\frac{8 π ^{4}}{3}

Decimal

259.75757 \dots

Solution steps

\int_{0}^{2 π} \frac{2 θ ^{3}}{3} d θ

Take the constant out:

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

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

Apply the Power Rule

: [\frac{θ ^{4}}{4}]_{0}^{2 π}

= \frac{2}{3} [\frac{θ ^{4}}{4}]_{0}^{2 π}

Compute the boundaries:

4 π^{4}

= \frac{2}{3} \cdot 4 π^{4}

Simplify

= \frac{8 π ^{4}}{3}

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What is the integral from 0 to 2pi of ((2θ^3)/3) ?
The integral from 0 to 2pi of ((2θ^3)/3) is (8pi^4)/3