Solve = ∫ (from - pi / 2 to pi / 2) of [3/4r^4]_0^2cos(θ)dθ

$$= \int _ { - \pi / 2 } ^ { \pi / 2 } [ \frac { 3 } { 4 } r ^ { 4 } ] _ { 0 } ^ { 2 \cos ( \theta ) } d \theta$$

Evaluate

$$\frac{3\pi r^{4}}{4}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{3r^{4}}{4}\mathrm{d}x$$

Find the integral of \frac{3r^{4}}{4} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3r^{4}}{4}x$$

Simplify.

$$\frac{3r^{4}x}{4}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{3}{4}r^{4}\times \frac{1}{2}\pi -\frac{3}{4}r^{4}\left(-1\right)\times \frac{1}{2}\pi$$

Simplify.

$$\frac{3r^{4}\pi }{4}$$

Differentiate w.r.t. r

$$3\pi r^{3}$$