Chapter 13: Q. 25 (page 1024)

Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ$

Short Answer

Expert verified

The integral's value is $2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ = \frac{π}{4}$

Step by step solution

given information

Let consider the given integral is $2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} r d r d θ$

Finding establish the expression

The goal of this issue is to sketch the region and assess the expression using polar coordinates $2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} r d r d θ$

Using, $r = 0, r = \sin θ and θ = 0, θ = π / 2$

$θ$	$r = \sin 3 θ$
0	0
$π / 6$	$1.0$
$π / 4$	$0.7071$
$π / 3$	$0$
$π / 2$	$- 1.0$

Calculations

$\begin{aligned} 2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ = 2 \int_{0}^{π / 2} {[\frac{r^{2}}{2}]}_{0}^{\sin 3 θ} dθ \\ = 2 \int_{0}^{π / 2} [\frac{\sin^{2} 3 θ - 0}{2}] θ \\ = 2 \int_{0}^{π / 2} [\frac{(1 - \cos 6 θ)}{4}] dθ \end{aligned}$

Integrate in relation to $θ$ ,

$2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ = 2 {[\frac{{\frac{(θ - (\sin 6 θ)}{6}}}{4}]}_{0}^{π / 2} [\int \cos xdx = \sin x]$

Pointing the limits,

$2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ = 2 [\frac{{\frac{π / 2 - (\sin 3 π)}{6} - 0}}{4}] = \frac{π}{4}$

As a result, the integral value is $2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ = \frac{π}{4}$

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Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ$

Short Answer

Step by step solution

given information

Finding establish the expression

Calculations

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Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.2∫0π/2 ∫0sin⁡3θ rdrdθ

Short Answer

Step by step solution

given information

Finding establish the expression

Calculations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

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Applied Mathematics

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Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 \int_{0}^{π / 2} \int_{0}^{\sin 3 θ} rdrdθ$