What is the integral from 0 to pi of 1/2 (2θ)^2 ?

The integral from 0 to pi of 1/2 (2θ)^2 is (2pi^3)/3

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integral from 0 to pi of 1/2 (2θ)^2

\int_{0}^{π} \frac{1}{2} (2 θ)^{2} d θ

\frac{2 π ^{3}}{3}

Decimal

20.67085 \dots

Solution steps

\int_{0}^{π} \frac{1}{2} (2 θ)^{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}^{π} (2 θ)^{2} d θ

Simplify

(2 θ)^{2} : 4 θ^{2}

= \frac{1}{2} \cdot \int_{0}^{π} 4 θ^{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 4 \cdot \int_{0}^{π} θ^{2} d θ

Apply the Power Rule

: [\frac{θ ^{3}}{3}]_{0}^{π}

= \frac{1}{2} \cdot 4 [\frac{θ ^{3}}{3}]_{0}^{π}

Simplify

\frac{1}{2} \cdot 4 [\frac{θ ^{3}}{3}]_{0}^{π} : 2 [\frac{θ ^{3}}{3}]_{0}^{π}

= 2 [\frac{θ ^{3}}{3}]_{0}^{π}

Compute the boundaries:

\frac{π ^{3}}{3}

= 2 \cdot \frac{π ^{3}}{3}

Simplify

= \frac{2 π ^{3}}{3}

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What is the integral from 0 to pi of 1/2 (2θ)^2 ?
The integral from 0 to pi of 1/2 (2θ)^2 is (2pi^3)/3