Explanation Given Information: The provided equations are y = 10 x ( 2 + 5 x ) , y = 0 , x = 1 and x = 2 . Formula used: The formula for the integral ∫ 1 u ( a + b u ) d u is, ∫ 1 u ( a + b u ) d u = 1 a ln | u a + b u | + C The General power differentiation Rule is; d d x [ u n ] = n u n − 1 d u d x Calculation: Consider u = x . Differentiate the considered function with respect to x using power rule of differentiation. d u = d x Consider the provided expression: y = 10 x ( 2 + 5 x ) To find the exact area of region bounded by the graphs of equations y = 10 x ( 2 + 5 x ) , y = 0 , x = 1 and x = 2 , integrate equation y = 10 x ( 2 + 5 x ) with lower limit x = 1 and higher limit x = 2 , which can be mathematically written as: ∫ 1 2 y d x = ∫ 1 2 10 x ( 2 + 5 x ) d x Rewrite. ∫ 1 2 y d x = 10 ∫ 1 2 1 x ( 2 + 5 x ) d x Here, a = 2 , b = 5 , u = x and d u = d x . Substitute u for x , a for 2 , b for 5 , and d u for d x . ∫ 1 2 y d x = 10 ∫ 1 2 1 u ( 2 + 5 u ) d x Use the above integration formula and solve the above integral as; ∫ 1 2 y d x = 10 [ 1 a ln | u a + b u | ] 1 2 Substitute x for u , 2 for a and 5 for b . ∫ 1 2 y d x = 10 [ 1 2 ln | x 2 + 5 x | ] 1 2 = 5 [ ln | x 2 + 5 x | ] 1 2 = 5 [ ln | 2 2 + 5 ( 2 ) | − ln | 1 2 + 5 ( 1 ) | ] = 5 [ ln | 2 12 | − ln | 1 7 | ] Further solve

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.2, Problem 48E

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To calculate: The exact area of region bounded by the graphs of equations y=10x(2+5x), y=0, x=1 and x=2.

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Chapter 6.2, Problem 47E

Chapter 6.2, Problem 49E

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Chapter 6 Solutions

Calculus: An Applied Approach (MindTap Course List)

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The exact area of region bounded by the graphs of equations y = 10 x ( 2 + 5 x ) , y = 0 , x = 1 and x = 2 .

Solution Summary: The author calculates the exact area of region bounded by the graphs of equations y=10x(2+5x

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To calculate: The exact area of region bounded by the graphs of equations y=10x(2+5x), y=0, x=1 and x=2.

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Chapter 6 Solutions