What is the integral from 0 to cos(θ) of r^3sin(θ) ?

The integral from 0 to cos(θ) of r^3sin(θ) is (cos^4(θ)sin(θ))/4

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integral from 0 to cos(θ) of r^3sin(θ)

\int_{0}^{c o s (θ)} r^{3} sin (θ) d r

\frac{cos ^{4} ( θ ) sin ( θ )}{4}

Solution steps

\int_{0}^{c o s (θ)} r^{3} sin (θ) d r

Take the constant out:

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

= sin (θ) \cdot \int_{0}^{c o s (θ)} r^{3} d r

Apply the Power Rule

: [\frac{r ^{4}}{4}]_{0}^{c o s (θ)}

= sin (θ) [\frac{r ^{4}}{4}]_{0}^{c o s (θ)}

Compute the boundaries:

\frac{cos ^{4} ( θ )}{4}

= sin (θ) \frac{cos ^{4} ( θ )}{4}

Simplify

= \frac{cos ^{4} ( θ ) sin ( θ )}{4}

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What is the integral from 0 to cos(θ) of r^3sin(θ) ?
The integral from 0 to cos(θ) of r^3sin(θ) is (cos^4(θ)sin(θ))/4