Question

Complete example 2 by evaluating the integral expression

${\int }_0^{\frac{2\pi }{3}}{(\frac{1}{2}+\cos \theta )}^{2}d\theta -{\int }_{\pi }^{\frac{4\pi }{3}}{(\frac{1}{2}+\cos \theta )}^{2}d\theta$

Short answer

The value of the integral is 2.08

Step-by-step solution

Given information

We are given an integral${\int }_0^{\frac{2\pi }{3}}{(\frac{1}{2}+\cos \theta )}^{2}d\theta -{\int }_{\pi }^{\frac{4\pi }{3}}{(\frac{1}{2}+\cos \theta )}^{2}d\theta$

Evaluate

We have,

$$\begin{gathered} {\int }_0^{\frac{2\pi }{3}}{(\frac{1}{2}+\cos \theta )}^{2}d\theta -{\int }_{\pi }^{\frac{4\pi }{3}}{(\frac{1}{2}+\cos \theta )}^{2}d\theta \\ ={\int }_0^{\frac{2\pi }{3}}(\frac{1}{4}+\cos \theta +{\cos }^2\theta )d\theta - \\ {\int }_{\pi }^{\frac{4\pi }{3}}(\frac{1}{4}+\cos \theta +{\cos }^2\theta )d\theta \\ ={{[\frac{1}{4}\theta +\sin \theta +\frac{1}{2}\theta +\frac{\sin 2\theta }{4}]}^{\frac{2\pi }{3}}}_0 \\ -{{[\frac{1}{4}\theta +\sin \theta +\frac{1}{2}\theta +\frac{\sin 2\theta }{4}]}^{\frac{4\pi }{3}}}_{\pi } \\ =2.22-0.1358=2.08 \end{gathered}$$