Chapter 13: Q. 22 (page 1027)

Each of the integrals or integral expressions in Exercises represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 \int_{0}^{\frac{π}{2}} \int_{0}^{\sin θ} rdrdθ$

Short Answer

Expert verified

The value of integer is $2 \int_{0}^{\frac{π}{2}} \int_{0}^{\sin θ} rdrdθ = \frac{π}{4}$

Step by step solution

Given information

The function is $2 \int_{0}^{\frac{π}{2}} \int_{0}^{\sin θ} rdrdθ$

Calculation of integrals

The goal of this issue is to sketch the region and assess the expression using polar coordinates $2 \int_{0}^{π} \int_{0}^{1 + \cos θ} rdrdθ$

using the values , $r = 0, r = 1 + \cos θ and θ = 0, θ = π$

Use the table above to draw the region $2 \int_{0}^{π} \int_{0}^{1 + \cos θ} rdrdθ = 2 \int_{0}^{π} {[\frac{r^{2}}{2}]}_{0}^{1 + \cos θ} dθ = 2 \int_{0}^{π} [\frac{(1 + \cos θ)^{2} - 0}{2}] dθ = 2 \int_{0}^{π} [\frac{(1 + 2 \cos θ + \cos^{2} θ)}{2}] dθ [(a + b)^{2} = a^{2} + 2 ab + b^{2}] = 2 \int_{0}^{π} [\frac{\{1 + 2 \cos θ + \frac{(1 + \cos 2 θ)}{2}\}}{2}] dθ = 2 \int_{0}^{x} [\frac{3 + 4 \cos θ + \cos 2 θ}{4}] dθ$

We can integrating in terms of $θ$

$2 \int_{0}^{π} \int_{0}^{1 + \cos θ} rdrdθ = 2 {[\frac{3 θ + 4 \sin θ + \frac{(\sin 2 θ)}{2}}{4}]}_{0}^{π}$

$[\int \sin xdx = - \cos x, \int \cos xdx = \sin x]$

Substituting the limits of derivatives,

$2 \int_{0}^{x} \int_{0}^{1 \cos θ} rdrdθ = 2 [\frac{3 π + 4 \sin π + \frac{(\sin 2 π)}{2} - 0}{4}]$

As a result, the integral value is $\int_{0}^{2 π} \int_{0}^{1 + \sin θ} rdrdθ = \frac{3 π}{2}$

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Each of the integrals or integral expressions in Exercises represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 \int_{0}^{\frac{π}{2}} \int_{0}^{\sin θ} rdrdθ$

Short Answer

Step by step solution

Given information

Calculation of integrals

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Each of the integrals or integral expressions in Exercises represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.2∫0π2 ∫0sin⁡θ rdrdθ

Short Answer

Step by step solution

Given information

Calculation of integrals

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Each of the integrals or integral expressions in Exercises represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 \int_{0}^{\frac{π}{2}} \int_{0}^{\sin θ} rdrdθ$