Explanation Given: The result of Exercise 76 is ∫ a b f ( x ) d x = ∫ a p f ( x ) d x + ∫ p b f ( x ) d x . Calculation: From the given piecewise function , consider f ( x ) = 2 x if 1 ≤ x < 4 . According to the result of Exercise 76, a = 1 , p = 4 . Therefore, it can be written as in terms of integral ∫ 1 4 2 x d x . Similarly, the function f ( x ) = 10 − 2 x if 4 ≤ x ≤ 6 can be expressed as ∫ 4 6 ( 10 − 2 x ) d x . Then by the result of Exercise 76, ∫ 1 6 f ( x ) d x = ∫ 1 4 2 x d x + ∫ 4 6 ( 10 − 2 x ) d x (1) Use online graphing calculator and obtain the graph y = 2 x on the interval [ 1 , 4 ) as shown in Figure 1. From Figure 1, it is observed that the graph represents the trapezoid of height 2 and the parallel sides 8 and 2 respectively. The area of the trapezoid is, ∫ 1 4 2 x d x = 1 2 h ( a + b ) = 1 2 ( 3 ) ( 8 + 2 ) = 1 2 3 ( 10 ) = 15 Thus, the net area of the region over the interval [ 1 , 4 ) between y = 2 x and the x -axis is 15

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 5.2, Problem 78E

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To evaluate: The value of ∫16f(x)dx , where f(x)={2x if 1≤x<410−2x if 4≤x≤6 by using the result of exercise 76 and geometry.

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Chapter 5.2, Problem 77E

Chapter 5.2, Problem 79E

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The value of ∫ 1 6 f ( x ) d x , where f ( x ) = { 2 x if 1 ≤ x < 4 10 − 2 x if 4 ≤ x ≤ 6 by using the result of exercise 76 and geometry.

Solution Summary: The author evaluates the value of displaystyle 'underset' by using the result of Exercise 76 and geometry.

Videos

To evaluate: The value of ∫16f(x)dx , where f(x)={2x if 1≤x<410−2x if 4≤x≤6 by using the result of exercise 76 and geometry.

Want to see the full answer?

Chapter 5 Solutions

The value of ∫ 1 6 f ( x ) d x , where f ( x ) = { 2 x if 1 ≤ x &lt; 4 10 − 2 x if 4 ≤ x ≤ 6 by using the result of exercise 76 and geometry.

Solution Summary: The author evaluates the value of displaystyle 'underset' by using the result of Exercise 76 and geometry.

Videos

To evaluate: The value of ∫16f(x)dx , where f(x)={2x if 1≤x<410−2x if 4≤x≤6 by using the result of exercise 76 and geometry.

Want to see the full answer?

Chapter 5 Solutions

The value of ∫ 1 6 f ( x ) d x , where f ( x ) = { 2 x if 1 ≤ x < 4 10 − 2 x if 4 ≤ x ≤ 6 by using the result of exercise 76 and geometry.