Explanation Given: The two curves has a function of y = x x 2 + 1 and y = x 2 x 3 + 1 . Calculation: Let y = f ( x ) , then f ( x ) = x x 2 + 1 and y = g ( x ) , then g ( x ) = x 2 x 3 + 1 . The intersection points of the curves can calculate by equating the curves as shown below: f ( x ) = g ( x ) (1) Here, the top curve function is f ( x ) and the bottom curve function is g ( x ) . x x 2 + 1 = x 2 x 3 + 1 x 2 + 1 = x x 3 + 1 x 2 + 1 = x 5 + x 2 x 5 = 1 x = 1 Draw the graph for the function y = x x 2 + 1 and y = x 2 x 3 + 1 by substituting different values for x . Shade the region lies between the intersecting points of the curves. The region enclosed by the curves y = x x 2 + 1 and y = x 2 x 3 + 1 is shown in Figure 1. Refer to Figure 1 to observe that the shaded region is the region enclosed by the given curves. The curves are bounded by the top and bottom curve. Hence, the integration can be done with respect to x . From the graph the intersecting points of the curve are x = 0 and x = 1 . Find the area of the region bounded by the curves using the relation: A = ∫ a b ( f ( x ) − g ( x ) ) d x (2) Where, the lower limit is a and the upper limit is b . Substitute x x 2 + 1 for f ( x ) , x 2 x 3 + 1 for g ( x ) , 0 for a , and 1 for b in Equation (2). A = ∫ 0 1 ( x x 2 + 1 − x 2 x 3 + 1 ) d x = ∫ 0 1 ( x x 2 + 1 ) d x − ∫ 0 1 ( x 2 x 3 + 1 ) d x Let u = 1 + x 2 and take derivative of u with respect to x . d u d x = d d x ( 1 + x 2 ) = 2 x d u 2 = x d x Since the variable d x changed into d u , the limit of the integration change as follows, Since x = 1 , then u = 2 and x = 0 , then u = 1

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ISBN: 9780100850668

Author: Stewart

Publisher: YUZU

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Chapter 5.1, Problem 32E

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To draw: The region enclosed by the given curves and find its area of shaded region.

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Chapter 5.1, Problem 31E

Chapter 5.1, Problem 33E

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Chapter 5 Solutions

EBK SINGLE VARIABLE CALCULUS, VOLUME 1

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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To draw: The region enclosed by the given curves and find its area of shaded region.

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To draw: The region enclosed by the given curves and find its area of shaded region.

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Chapter 5 Solutions