Explanation Given: The curve function is y = x 2 (1) Calculation: Differentiate Equation (1) with respect to x . d y d x = 2 x (2) Find the slope of the tangent ( d y d x ) using the coordinates of the line ( 1 , 1 ) , as shown below: Substitute 1 for x in Equation (2). d y d x = 2 ( 1 ) = 2 The general expression for the equation of a line is shown below. y − y 1 = d y d x ( x − x 1 ) (3) Here, the coordinates of the tangent line are ( x 1 , y 1 ) and the slope is d y d x . Substitute 1 for y 1 , 2 for d y d x , and 1 for x 1 in Equation (3). y − 1 = 2 ( x − 1 ) y = 2 x − 2 + 1 y = 2 x − 1 (4) The procedure to draw the graph for the curve functions y = x 2 and y = 2 x − 1 is shown below: Draw the graph for the function y = x 2 by substituting different values for x. Similarly plot the function y = 2 x − 1 by substituting different values for x . The region enclosed by the curves y = x 2 and y = 2 x − 1 is shown in Figure 1. Refer to Figure 1. The limits to integrate the function with respect to y is x = 0 to x = 1 . Rearrange Equation (1), as shown below. x = y Rearrange Equation (4), as shown below. 2 x = y + 1 x = 1 2 ( y + 1 ) The expression to find the area enclosed by the parabola and the line is shown below. A = ∫ a b [ f ( y ) − g ( y ) ] d y (5) Here, the right curve function is f ( y ) , the left curve function is g ( y ) , the lower limit is a , and the upper limit is b

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ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5.1, Problem 56E

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To find: The area of the region enclosed by the parabola y=x2 and the tangent line (1, 1).

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Chapter 5.1, Problem 55E

Chapter 5.1, Problem 57E

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Chapter 5 Solutions

Single Variable Calculus

Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 6E Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 8E Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....

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The area of the region enclosed by the parabola y = x 2 and the tangent line ( 1 , 1 ) .

Solution Summary: The author explains how to find the region enclosed by the parabola and the tangent line.

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To find: The area of the region enclosed by the parabola y=x2 and the tangent line (1, 1).

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