Solve ∫ (from 0 to 2 pi) of 1/2x^2sin(θ)dθ

$$\int_{ 0 }^{ 2 \pi } \frac{ 1 }{ 2 } { x }^{ 2 } \sin ( \theta ) d \theta$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result.

$$-\frac{x^{2}}{2}\cos(\theta )$$

Simplify.

$$-\frac{x^{2}\cos(\theta )}{2}$$

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.

$$-\frac{1}{2}x^{2}\cos(2\pi )+\frac{1}{2}x^{2}\cos(0)$$

Simplify.

$$0$$

Differentiate w.r.t. x

$$0$$