Solve ∫ (from 0 to pi / 3) of (1-cosθ)^2

$$\int _ { 0 } ^ { \pi / 3 } ( 1 - \cos \theta ) ^ { 2 }$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

$$\int \left(1-\cos(\theta )\right)^{2}\mathrm{d}x$$

Find the integral of \left(1-\cos(\theta )\right)^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(1-\cos(\theta )\right)^{2}x$$

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.

$$\frac{1}{3}\left(1-\cos(\theta )\right)^{2}\pi +0\left(1-\cos(\theta )\right)^{2}$$

Simplify.

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

Differentiate w.r.t. θ

$$-\frac{2\pi \sin(\theta )\left(\cos(\theta )-1\right)}{3}$$