Solve ∫ (from 0 to 2) of R^2sin(θ)dR

$$\int_{ 0 }^{ 2 } { R }^{ 2 } \sin ( \theta ) d R$$

Evaluate

$$\frac{8\sin(\theta )}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int R^{2}\sin(\theta )\mathrm{d}R$$

Factor out the constant using \int af\left(R\right)\mathrm{d}R=a\int f\left(R\right)\mathrm{d}R.

$$\sin(\theta )\int R^{2}\mathrm{d}R$$

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}.

$$\sin(\theta )\times \frac{R^{3}}{3}$$

Simplify.

$$\frac{\sin(\theta )R^{3}}{3}$$

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.

$$\frac{1}{3}\sin(\theta )\times 2^{3}-\frac{1}{3}\sin(\theta )\times 0^{3}$$

Simplify.

$$\frac{8\sin(\theta )}{3}$$

Differentiate w.r.t. θ

$$\frac{8\cos(\theta )}{3}$$