Explanation Given information: The Equation of the curves y = 1 + x 4 and y = 3 − x 3 . Calculation: Show the Equation of the curves as follows: y = 2 x − x 2 3 (1) y = x 2 x 2 + 1 (2) Procedure to sketch the region bounded by the two curves is explained below: Draw the graph for the function y = 2 x − x 2 3 and y = x 2 x 2 + 1 by substituting different values for x . Sketch the region enclosed by the curves as shown in Figure 1. From the Figure 1, the intersection points are x = 0 and x = 1.75 . Hence, the approximate x- coordinates of the point of intersection of the given curves are x = 0 and x = 1.75 . Consider that the outer radius of the region r 1 = 2 x − x 2 3 and the inner radius of the region r 2 = x 2 x 2 + 1 within the limit − 1 to 0.857 . Then the Area enclosed by the curves is as shown below. A ( x ) = π r 1 2 − π r 2 2 = π ( 2 x − x 2 3 ) 2 − π ( x 2 x 2 + 1 ) 2 Thus, the volume of the solid obtained by rotating about the x -axis as shown below. V = ∫ a b A ( x ) d x (3) Substitute 0 for a , 1

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ISBN: 9781305266636

Author: James Stewart

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Chapter 5.2, Problem 36E

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To find: The approximate x-coordinates of the point of intersection of the given curves and the volume of the solid obtained by rotating about the x-axis.

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Chapter 5.2, Problem 35E

Chapter 5.2, Problem 37E

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Use any method to find the volume, V, of the solid obtained by rotating the region enclosed by the curves about the given axis. y² = x¯¹, x = 1, (Express numbers in exact form. Use symbolic notation and fractions where needed.) X = 6, axis y = −6 V = (156+24√√6 +ln(6))π Incorrect < Feedback You have not correctly calculated the volume of the solid. Slice the region vertically and use the washer method to find the volume of the solid of revolution. b [*(Router - Riner) dx a V = π X Note that a and b are the x-coordinates of the intersection points. Find Router as the distance from the axis of rotation to the top point of the region. Find Rinner as the distance from the axis of rotation to the bottom point of the region.

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the infinite region in the first quadrantbetween the curve y = e-x and the x-axis. Find the volume of the solid generated by revolving the region about the x-axis.

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The approximate x- coordinates of the point of intersection of the given curves and the volume of the solid obtained by rotating about the x -axis.

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To find: The approximate x-coordinates of the point of intersection of the given curves and the volume of the solid obtained by rotating about the x-axis.

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Chapter 5 Solutions