Chapter 9: Q. 40 (page 721)

Complete the calculation in Example 7 by using the trigonometric identity $\sin^{2} (\frac{θ}{2}) = \frac{1}{2} (1 - \cos θ)$ to show that $\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ = 4 \sqrt{2}$ .

Short Answer

Expert verified

The integral $\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ$ is equals to $4 \sqrt{2}$

Step by step solution

Given information

The integral is $\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ$

Calculation

Consider the integral, $\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ$ .

The objective is to solve the integral within the given limits.

By using the trigonometric identity the integral can be written as follows.

Take the integral,

$\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ = \int_{0}^{2 π} \sqrt{2 \sin^{2} \frac{θ}{2}} d θ [since \sin^{2} \frac{θ}{2} = \frac{1}{2} \sqrt{1 - \cos θ} \Rightarrow 2 \sin^{2} \frac{θ}{2} = \sqrt{1 - \cos θ}] = \sqrt{2} \int_{0}^{2 π} \sqrt{\sin^{2} \frac{θ}{2}} d θ = \sqrt{2} \int_{0}^{2 π} \sin \frac{θ}{2} d θ$

Thus,

$\int_{0}^{2 π} \sqrt{2 \sin^{2} \frac{θ}{2}} d θ = \sqrt{2} {(- 2 \cos \frac{θ}{2})}_{0}^{2 π} [since \int_{0}^{2 π} \sin \frac{θ}{2} d θ = {(- 2 \cos \frac{θ}{2})}_{0}^{2 π}] = \sqrt{2} (- 2 \cos \frac{2 π}{2} + 2 \cos \frac{0}{2}) By applying the limits = \sqrt{2} (- 2 \cos π + 2 \cos \frac{0}{2}) = \sqrt{2} (- 2 (- 1) + 1 \cdot 2)$

On further simplification,

$\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ = \sqrt{2} (2 + 2) = \sqrt{2} \times 4 \int_{0}^{2 π} \sqrt{1 - \cos θ} d θ = 4 \sqrt{2}$

Therefore, the integral $\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ$ is equals to $4 \sqrt{2}$ .

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Complete the calculation in Example 7 by using the trigonometric identity $\sin^{2} (\frac{θ}{2}) = \frac{1}{2} (1 - \cos θ)$ to show that $\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ = 4 \sqrt{2}$ .

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Complete the calculation in Example 7 by using the trigonometric identity sin2θ2=12(1-cosθ)to show that ∫02π1-cosθdθ=42.

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Complete the calculation in Example 7 by using the trigonometric identity $\sin^{2} (\frac{θ}{2}) = \frac{1}{2} (1 - \cos θ)$ to show that $\int_{0}^{2 π} \sqrt{1 - \cos θ} d θ = 4 \sqrt{2}$ .