Find he asca of Mylies btwen inside borhThascurrsr-1-Sino and r I + Sin

Name: Find he asca of Mylies btwen inside borhThascurrsr-1-Sino and r I + S
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Description: Find he asca of Mylies btwen inside borhThascurrsr-1-Sino and r I + Sin

Answered: Find Mhe wiea of My reegon That ^Thas…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find he asca of My
lies btwen inside borh
Thas
currs
r-1-
Sino and r I + Sin

Expert Solution

Step 1

Given :

The curves $r = 1 - \sin θ$ and $r = 1 + \sin θ$ .

To determine :

The area of the region that lies between the curves $r = 1 - \sin θ$ and $r = 1 + \sin θ$ .

Step 2

Since, $r = 1 - \sin θ$ and $r = 1 + \sin θ$ .

Since, we need to find the area for the two loop as shown in the figure and the two loops are symmetric,

Let us start with one loop and find the area.

Area of the loop 1 = $\frac{1}{2} \int_{θ_{1}}^{θ_{2}} r^{2} d θ$

For the first loop, it is enclosed by the two curves (horizontally along x axis). So we have $0 \to \frac{π}{2}$ ( blue curve ) and $\frac{π}{2} \to π$ ( red curve ) .

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