Explanation Given : f ( x , y ) = y ; x 2 + y 2 ≤ 1 ; ( x − 1 ) 2 + y 2 ≤ 1 Formula Used: Change of Variables formula Calculation : Step 1: D is the common region of the two circles shown in the figure. To evaluate the integral we decompose D into three regions D1, D2, and D3 as shown in the figure. Thus, ∬ D y d A = ∬ D 1 y d A + ∬ D 2 y d A + ∬ D 3 y d A Step 2: We compute each integral separately. D 1 : The domain D 1 lies in the angular sector − π 3 ≤ θ ≤ π 3 (where π 3 = tan − 1 3 2 1 2 = tan − 1 3 ). The circle x 2 + y 2 = 1 , has polar equation r = 1 , therefore D1 has the following definition: D 1 : − π 3 ≤ θ ≤ π 3 , 0 ≤ r ≤ 1 Step 3: We use Change of Variables formula in the integral to write ∬ D 1 y d A = ∫ − π 3 π 3 ∫ 0 1 ( r sin θ ) r d r d θ = ∫ − π 3 π 3 ∫ 0 1 r 2 sin θ d r d θ = ∫ − π 3 π 3 ∫ 0 1 r 3 sin θ 3 | r = 0 1 d θ = ∫ − π 3 π 3 ∫ 0 1 sin θ 3 d θ = − cos θ 3 | − π 3 π 3 d θ = 0 Step 4: The angle θ is changing in D2 from π 3 to π 2 . The circle ( x − 1 ) 2 + y 2 = 1 has polar equation r = 2 c o s θ . Therefore, D 2 has the following description: D 2 : π 3 ≤ θ ≤ π 2 , 0 ≤ r ≤ 2 cos θ

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, Problem 20E

To determine

To calculate: The integral over the region $f (x, y) = y; x^{2} + y^{2} \leq 1; {(x - 1)}^{2} + y^{2} \leq 1$ by changing to polar coordinates.

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Chapter 16.4, Problem 19E

Chapter 16.4, Problem 21E

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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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The integral over the region f ( x , y ) = y ; x 2 + y 2 ≤ 1 ; ( x − 1 ) 2 + y 2 ≤ 1 by changing to polar coordinates.

Solution Summary: The author explains how to calculate the integral over the region f (x, y) by changing to polar coordinates.

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To calculate: The integral over the region $f (x, y) = y; x^{2} + y^{2} \leq 1; {(x - 1)}^{2} + y^{2} \leq 1$ by changing to polar coordinates.

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Chapter 16 Solutions

The integral over the region f ( x , y ) = y ; x 2 + y 2 ≤ 1 ; ( x − 1 ) 2 + y 2 ≤ 1 by changing to polar coordinates.

Solution Summary: The author explains how to calculate the integral over the region f (x, y) by changing to polar coordinates.

Videos

To calculate: The integral over the region f(x,y)=y;x2+y2≤1;(x−1)2+y2≤1 by changing to polar coordinates.

Want to see the full answer?

Chapter 16 Solutions

To calculate: The integral over the region $f (x, y) = y; x^{2} + y^{2} \leq 1; {(x - 1)}^{2} + y^{2} \leq 1$ by changing to polar coordinates.