Solve ∫ (from 2 cos θ to 4 cos θ) of r ^ 2 d r

$$\int _ { 2 \cos \theta } ^ { 4 \cos \theta } r ^ { 2 } d r$$

Evaluate

$$\frac{56\left(\cos(\theta )\right)^{3}}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int r^{2}\mathrm{d}r$$

Since \int r^{k}\mathrm{d}r=\frac{r^{k+1}}{k+1} for k\neq -1, replace \int r^{2}\mathrm{d}r with \frac{r^{3}}{3}.

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

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.

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

Simplify.

$$\frac{56\left(\cos(\theta )\right)^{3}}{3}$$

Differentiate w.r.t. θ

$$-56\sin(\theta )\left(\cos(\theta )\right)^{2}$$