Chapter 9: Q. 35 (page 775)

Areas of regions bounded by polar functions: Find the areas of the following regions.
The area bounded by $r = 1 - \sin θ$

Short Answer

Expert verified

The area bounded by the function is $A = π {unit}^{2}$

Step by step solution

Given information

Given curve is $r = 1 - \sin θ$

Area

The area bounded by the curve is,

$A = \frac{1}{2} \int_{α}^{β} r^{2} d θ A = \frac{1}{2} \int_{0}^{2 π} {(1 - \sin θ)}^{2} d θ A = \frac{1}{2} \int_{0}^{2 π} (1 + \sin^{2} θ - 2 \sin θ) d θ A = \frac{1}{2} \int_{0}^{2 π} \frac{(3 - \cos 2 θ)}{2} - 2 \sin θ d θ A = \frac{1}{2} {[\frac{3 θ}{2} - \frac{\sin 2 θ}{4} + 2 \cos θ]}_{0}^{2 π} A = \frac{1}{2} [\frac{6 π}{3} - \frac{\sin 4 π}{4} + 2 \cos 2 π - (0 + 2 \cos 0)] A = \frac{1}{2} [2 π - 0 + 2 - 2] A = π {unit}^{2}$

The area bounded by the function is $A = π {unit}^{2}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Areas of regions bounded by polar functions: Find the areas of the following regions.
The area bounded by $r = 1 - \sin θ$

Short Answer

Step by step solution

Given information

Area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Logic and Functions

Calculus

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Areas of regions bounded by polar functions: Find the areas of the following regions.The area bounded by r=1−sinθ

Short Answer

Step by step solution

Given information

Area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Logic and Functions

Calculus

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Areas of regions bounded by polar functions: Find the areas of the following regions.
The area bounded by $r = 1 - \sin θ$