Chapter 9: Q. 37 (page 756)

Complete example 2 by evaluating the integral expression
$\int_{0}^{\frac{2 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ - \int_{π}^{\frac{4 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ$

Short Answer

Expert verified

The value of the integral is 2.08

Step by step solution

Given information

We are given an integral $\int_{0}^{\frac{2 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ - \int_{π}^{\frac{4 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ$

Evaluate

We have,

$\int_{0}^{\frac{2 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ - \int_{π}^{\frac{4 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ = \int_{0}^{\frac{2 π}{3}} (\frac{1}{4} + \cos θ + \cos^{2} θ) d θ - \int_{π}^{\frac{4 π}{3}} (\frac{1}{4} + \cos θ + \cos^{2} θ) d θ = {[\frac{1}{4} θ + \sin θ + \frac{1}{2} θ + \frac{\sin 2 θ}{4}]}^{\frac{2 π}{3}}_{0} - {[\frac{1}{4} θ + \sin θ + \frac{1}{2} θ + \frac{\sin 2 θ}{4}]}^{\frac{4 π}{3}}_{π} = 2.22 - 0.1358 = 2.08$

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Complete example 2 by evaluating the integral expression
$\int_{0}^{\frac{2 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ - \int_{π}^{\frac{4 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ$

Short Answer

Step by step solution

Given information

Evaluate

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Complete example 2 by evaluating the integral expression∫02π3(12+cosθ)2dθ-∫π4π3(12+cosθ)2dθ

Short Answer

Step by step solution

Given information

Evaluate

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Complete example 2 by evaluating the integral expression
$\int_{0}^{\frac{2 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ - \int_{π}^{\frac{4 π}{3}} {(\frac{1}{2} + \cos θ)}^{2} d θ$