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 y = c − x 2 -------- (i) y = x 2 − c ------- (ii) Trace the curves on the X Y plane. Find the point of intersection of the curves. Substitute (ii) in (i) x 2 − c = c − x 2 x 2 + x 2 = c + c 2 x 2 = 2 c x 2 = c x 2 − c = 0 x 2 − ( c ) 2 = 0 ( x + c ) ( x − c ) = 0 , [Using the formula A 2 − B 2 = ( A + B ) ( A − B ) ] x + c = 0 or x − c = 0 x = − c or x = c Substitute x = − c in (i) y = c − x 2 y = c − ( − c ) 2 = c − c = 0 ( − c , 0 ) is one point. Substitute x = c in (i) y = c − x 2 = c − ( c ) 2 = c − c = 0 ( c , 0 ) is another point. ( − c , 0 ) , ( c , 0 ) are the points of intersection of the curves (i) and (ii) The region of integration is A B C D A Here the curve y = c − x 2 is the upper curve and y = x 2 − c is lower curveand x = − c , x = c are vertical elements. Area of A B C D A = 2 [Area of B C D B ] ------- (iii) Formula used: ∫ x n d x = x n + 1 n + 1 ∫ 1 d x = x ( A m ) n = A m n Area of B C D B = ∫ 0 c [ ( c − x 2 ) − ( x 2 − c ) ] d x...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.1, Problem 53E

To determine

The area enclosed by the curves as function of $c$ .

For what value of $c$ , area is $1$

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Chapter 6.1, Problem 52E

Chapter 6.1, Problem 54E

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Chapter 6 Solutions

Calculus - Standalone book

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The area enclosed by the curves as function of c . For what value of c , area is 1

Solution Summary: The author calculates the area enclosed by the curves as function of c by subtracting the lower curve from the upper curve and evaluating the integral.

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The area enclosed by the curves as function of $c$ .

For what value of $c$ , area is $1$

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The area enclosed by the curves as function of c . For what value of c , area is 1

Solution Summary: The author calculates the area enclosed by the curves as function of c by subtracting the lower curve from the upper curve and evaluating the integral.

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The area enclosed by the curves as function of c. For what value of c, area is 1

Want to see the full answer?

Chapter 6 Solutions

The area enclosed by the curves as function of $c$ .

For what value of $c$ , area is $1$