Explanation When area is enclosed by two curves y = f ( x ) and y = g ( x ) area is calculated using vertical elements by subtracting the lower curve from the upper curve and evaluating the integral. Given: The curves x 2 + y 2 = 2 ----------- (i) x 2 + ( y − 1 ) 2 = 1 ------ (ii) Place the curves on X Y plane with the scale X Maximum 4 X Minimum -4 X Scale 1 Y Maximum 3 Y Minimum -3 Y Scale 1 The region OABCO is the area of enclosed between the circles (i) & (ii) Point of intersection of the curves (i) & (ii) Formula used: ( A − B ) 2 = A 2 − 2 A B + B 2 A 2 − B 2 = ( A + B ) ( A − B ) From (i), x 2 = 2 − y 2 substitute in (ii) 2 − y 2 + ( y − 1 ) 2 = 1 2 − y 2 + y 2 − 2 y + 1 = 1 3 − 2 y = 1 3 − 1 = 2 y 2 = 2 y y = 1 Put y = 1 in (i) x 2 + y 2 = 2 x 2 + 1 = 2 x 2 + 1 − 2 = 0 x 2 − 1 = 0 ( x + 1 ) ( x − 1 ) = 0 x + 1 =0; x − 1 =0 x = − 1 ; x = 1 ∴ ( − 1 , 1 ) , ( 1 , 1 ) are the points of intersection...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 6.1, Problem 54E

To determine

The integral that represents the area between the circles $x^{2} + y^{2} = 2$ and $x^{2} + {(y - 1)}^{2} = 1$

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Chapter 6.1, Problem 53E

Chapter 6.1, Problem 55E

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Chapter 6 Solutions

Calculus - Standalone book

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Double and Triple Integrals; Author: Professor Dave Explains;https://www.youtube.com/watch?v=UubU3U2C8WM;License: Standard YouTube License, CC-BY

The integral that represents the area between the circles x 2 + y 2 = 2 and x 2 + ( y − 1 ) 2 = 1

Solution Summary: The author calculates the integral that represents the area between the circles by subtracting the lower curve from the upper curve and evaluating it.

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The integral that represents the area between the circles $x^{2} + y^{2} = 2$ and $x^{2} + {(y - 1)}^{2} = 1$

Want to see the full answer?

Chapter 6 Solutions

The integral that represents the area between the circles x 2 + y 2 = 2 and x 2 + ( y − 1 ) 2 = 1

Solution Summary: The author calculates the integral that represents the area between the circles by subtracting the lower curve from the upper curve and evaluating it.

Videos

The integral that represents the area between the circles x2+y2=2 and x2+(y−1)2=1

Want to see the full answer?

Chapter 6 Solutions

The integral that represents the area between the circles $x^{2} + y^{2} = 2$ and $x^{2} + {(y - 1)}^{2} = 1$