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Math Calculus Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

We need to prove that, for r < s , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where C is the point on the parabola at which the tangent line is parallel to the secant line A E ¯ . (a) We need to show that C has x coordinate ( r + s ) / 2 . (b) We need to prove that ABDE has area ( s − r ) 3 4 by viewing it as a parallelogram of height s − r and base of length C F ¯ . (c) We need to show that the triangle ACE has area ( s − r ) 3 8 by observing it has the same height and base as the parallelogram. (d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

We need to prove that, for r < s , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where C is the point on the parabola at which the tangent line is parallel to the secant line A E ¯ . (a) We need to show that C has x coordinate ( r + s ) / 2 . (b) We need to prove that ABDE has area ( s − r ) 3 4 by viewing it as a parallelogram of height s − r and base of length C F ¯ . (c) We need to show that the triangle ACE has area ( s − r ) 3 8 by observing it has the same height and base as the parallelogram. (d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

3rd Edition

ISBN: 9781319019846

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

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Question

Chapter 5.4, Problem 63E

To determine

We need to prove that, for $r < s$ , the area of the shaded region in Figure 9 is equal to the four-thirds the area of triangle ACE, where $C$ is the point on the parabola at which the tangent line is parallel to the secant line $\bar{A E}$ .

(a) We need to show that $C$ has x coordinate $(r + s) / 2$ .

(b) We need to prove that ABDE has area $\frac{{(s - r)}^{3}}{4}$ by viewing it as a parallelogram of height $s - r$ and base of length $\bar{C F}$ .

(c) We need to show that the triangle ACE has area $\frac{{(s - r)}^{3}}{8}$ by observing it has the same height and base as the parallelogram.

(d) We need to compute the shaded area as the area under the graph minus the area of a trapezoid. We need to prove the Archimedes’ result.

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Chapter 5.4, Problem 62E

Chapter 5.4, Problem 64E

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Chapter 5 Solutions

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - Prob. 49E Ch. 5.1 - Prob. 50E Ch. 5.1 - Prob. 51E Ch. 5.1 - Prob. 52E Ch. 5.1 - Prob. 53E Ch. 5.1 - Prob. 54E Ch. 5.1 - Prob. 55E Ch. 5.1 - 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Prob. 2E Ch. 5.8 - Prob. 3E Ch. 5.8 - Prob. 4E Ch. 5.8 - Prob. 5E Ch. 5.8 - Prob. 6E Ch. 5.8 - Prob. 7E Ch. 5.8 - Prob. 8E Ch. 5.8 - Prob. 9E Ch. 5.8 - Prob. 10E Ch. 5.8 - Prob. 11E Ch. 5.8 - Prob. 12E Ch. 5.8 - Prob. 13E Ch. 5.8 - Prob. 14E Ch. 5.8 - Prob. 15E Ch. 5.8 - Prob. 16E Ch. 5.8 - Prob. 17E Ch. 5.8 - Prob. 18E Ch. 5.8 - Prob. 19E Ch. 5.8 - Prob. 20E Ch. 5.8 - Prob. 21E Ch. 5.8 - Prob. 22E Ch. 5.8 - Prob. 23E Ch. 5.8 - Prob. 24E Ch. 5.8 - Prob. 25E Ch. 5.8 - Prob. 26E Ch. 5.8 - Prob. 27E Ch. 5.8 - Prob. 28E Ch. 5.8 - Prob. 29E Ch. 5.8 - Prob. 30E Ch. 5.8 - Prob. 31E Ch. 5.8 - Prob. 32E Ch. 5.8 - Prob. 33E Ch. 5.8 - Prob. 34E Ch. 5.8 - Prob. 35E Ch. 5.8 - Prob. 36E Ch. 5.8 - Prob. 37E Ch. 5.8 - Prob. 38E Ch. 5.8 - Prob. 39E Ch. 5.8 - Prob. 40E Ch. 5.8 - Prob. 41E Ch. 5.8 - Prob. 42E Ch. 5.8 - Prob. 43E Ch. 5.8 - Prob. 44E Ch. 5.8 - Prob. 45E Ch. 5.8 - Prob. 46E Ch. 5.8 - Prob. 47E Ch. 5.8 - Prob. 48E Ch. 5.8 - Prob. 49E Ch. 5.8 - Prob. 50E Ch. 5.8 - Prob. 51E Ch. 5.8 - Prob. 52E Ch. 5.8 - Prob. 53E Ch. 5.8 - Prob. 54E Ch. 5.8 - Prob. 55E Ch. 5.8 - Prob. 56E Ch. 5.8 - Prob. 57E Ch. 5.8 - Prob. 58E Ch. 5.8 - Prob. 59E Ch. 5.8 - Prob. 60E Ch. 5.8 - Prob. 61E Ch. 5.8 - Prob. 62E Ch. 5.8 - Prob. 63E Ch. 5.8 - Prob. 64E Ch. 5.8 - Prob. 65E Ch. 5.8 - Prob. 66E Ch. 5.8 - Prob. 67E Ch. 5.8 - Prob. 68E Ch. 5.8 - Prob. 69E Ch. 5.8 - Prob. 70E Ch. 5.8 - Prob. 71E Ch. 5.8 - Prob. 72E Ch. 5.8 - Prob. 73E Ch. 5.8 - Prob. 74E Ch. 5.8 - Prob. 75E Ch. 5.8 - Prob. 76E Ch. 5.8 - Prob. 77E Ch. 5.8 - Prob. 78E Ch. 5.8 - Prob. 79E Ch. 5.8 - Prob. 80E Ch. 5.8 - Prob. 81E Ch. 5.8 - Prob. 82E Ch. 5.8 - Prob. 83E Ch. 5.9 - Prob. 1PQ Ch. 5.9 - Prob. 2PQ Ch. 5.9 - Prob. 3PQ Ch. 5.9 - Prob. 4PQ Ch. 5.9 - Prob. 5PQ Ch. 5.9 - Prob. 1E Ch. 5.9 - Prob. 2E Ch. 5.9 - Prob. 3E Ch. 5.9 - Prob. 4E Ch. 5.9 - Prob. 5E Ch. 5.9 - Prob. 6E Ch. 5.9 - Prob. 7E Ch. 5.9 - Prob. 8E Ch. 5.9 - Prob. 9E Ch. 5.9 - Prob. 10E Ch. 5.9 - Prob. 11E Ch. 5.9 - Prob. 12E Ch. 5.9 - Prob. 13E Ch. 5.9 - Prob. 14E Ch. 5.9 - Prob. 15E Ch. 5.9 - Prob. 16E Ch. 5.9 - Prob. 17E Ch. 5.9 - Prob. 18E Ch. 5.9 - Prob. 19E Ch. 5.9 - Prob. 20E Ch. 5.9 - Prob. 21E Ch. 5.9 - Prob. 22E Ch. 5.9 - Prob. 23E Ch. 5.9 - Prob. 24E Ch. 5.9 - Prob. 25E Ch. 5.9 - Prob. 26E Ch. 5.9 - Prob. 27E Ch. 5.9 - Prob. 28E Ch. 5.9 - Prob. 29E Ch. 5.9 - Prob. 30E Ch. 5.9 - Prob. 31E Ch. 5.9 - Prob. 32E Ch. 5.9 - Prob. 33E Ch. 5.9 - Prob. 34E Ch. 5.9 - Prob. 35E Ch. 5.9 - Prob. 36E Ch. 5.9 - Prob. 37E Ch. 5.9 - Prob. 38E Ch. 5.9 - Prob. 39E Ch. 5.9 - Prob. 40E Ch. 5.9 - Prob. 41E Ch. 5.9 - Prob. 42E Ch. 5.9 - Prob. 43E Ch. 5.9 - Prob. 44E Ch. 5.9 - Prob. 45E Ch. 5.9 - Prob. 46E Ch. 5.9 - Prob. 47E Ch. 5.9 - Prob. 48E Ch. 5.9 - Prob. 49E Ch. 5.9 - Prob. 50E Ch. 5.9 - Prob. 51E Ch. 5.9 - Prob. 52E Ch. 5.9 - Prob. 53E Ch. 5.9 - Prob. 54E Ch. 5.9 - Prob. 55E Ch. 5.9 - Prob. 56E Ch. 5.9 - Prob. 57E Ch. 5.9 - Prob. 58E Ch. 5.9 - Prob. 59E Ch. 5.9 - Prob. 60E Ch. 5.9 - Prob. 61E Ch. 5.9 - Prob. 62E Ch. 5.9 - Prob. 63E Ch. 5 - Prob. 1CRE Ch. 5 - Prob. 2CRE Ch. 5 - Prob. 3CRE Ch. 5 - Prob. 4CRE Ch. 5 - Prob. 5CRE Ch. 5 - Prob. 6CRE Ch. 5 - Prob. 7CRE Ch. 5 - Prob. 8CRE Ch. 5 - Prob. 9CRE Ch. 5 - Prob. 10CRE Ch. 5 - Prob. 11CRE Ch. 5 - Prob. 12CRE Ch. 5 - Prob. 13CRE Ch. 5 - Prob. 14CRE Ch. 5 - Prob. 15CRE Ch. 5 - Prob. 16CRE Ch. 5 - Prob. 17CRE Ch. 5 - Prob. 18CRE Ch. 5 - Prob. 19CRE Ch. 5 - Prob. 20CRE Ch. 5 - Prob. 21CRE Ch. 5 - Prob. 22CRE Ch. 5 - Prob. 23CRE Ch. 5 - Prob. 24CRE Ch. 5 - Prob. 25CRE Ch. 5 - Prob. 26CRE Ch. 5 - Prob. 27CRE Ch. 5 - Prob. 28CRE Ch. 5 - Prob. 29CRE Ch. 5 - Prob. 30CRE Ch. 5 - Prob. 31CRE Ch. 5 - Prob. 32CRE Ch. 5 - Prob. 33CRE Ch. 5 - Prob. 34CRE Ch. 5 - Prob. 35CRE Ch. 5 - Prob. 36CRE Ch. 5 - Prob. 37CRE Ch. 5 - Prob. 38CRE Ch. 5 - Prob. 39CRE Ch. 5 - Prob. 40CRE Ch. 5 - Prob. 41CRE Ch. 5 - Prob. 42CRE Ch. 5 - Prob. 43CRE Ch. 5 - Prob. 44CRE Ch. 5 - Prob. 45CRE Ch. 5 - Prob. 46CRE Ch. 5 - Prob. 47CRE Ch. 5 - Prob. 48CRE Ch. 5 - Prob. 49CRE Ch. 5 - Prob. 50CRE Ch. 5 - Prob. 51CRE Ch. 5 - Prob. 52CRE Ch. 5 - Prob. 53CRE Ch. 5 - Prob. 54CRE Ch. 5 - Prob. 55CRE Ch. 5 - Prob. 56CRE Ch. 5 - Prob. 57CRE Ch. 5 - Prob. 58CRE Ch. 5 - Prob. 59CRE Ch. 5 - Prob. 60CRE Ch. 5 - Prob. 61CRE Ch. 5 - Prob. 62CRE Ch. 5 - Prob. 63CRE Ch. 5 - Prob. 64CRE Ch. 5 - Prob. 65CRE Ch. 5 - Prob. 66CRE Ch. 5 - Prob. 67CRE Ch. 5 - Prob. 68CRE Ch. 5 - Prob. 69CRE Ch. 5 - Prob. 70CRE Ch. 5 - Prob. 71CRE Ch. 5 - Prob. 72CRE Ch. 5 - Prob. 73CRE Ch. 5 - Prob. 74CRE Ch. 5 - Prob. 75CRE Ch. 5 - Prob. 76CRE Ch. 5 - Prob. 77CRE Ch. 5 - Prob. 78CRE Ch. 5 - Prob. 79CRE Ch. 5 - Prob. 80CRE Ch. 5 - Prob. 81CRE Ch. 5 - Prob. 82CRE Ch. 5 - Prob. 83CRE Ch. 5 - Prob. 84CRE Ch. 5 - Prob. 85CRE Ch. 5 - Prob. 86CRE Ch. 5 - Prob. 87CRE Ch. 5 - Prob. 88CRE Ch. 5 - Prob. 89CRE Ch. 5 - Prob. 90CRE Ch. 5 - Prob. 91CRE Ch. 5 - Prob. 92CRE Ch. 5 - Prob. 93CRE Ch. 5 - Prob. 94CRE Ch. 5 - Prob. 95CRE Ch. 5 - Prob. 96CRE Ch. 5 - Prob. 97CRE Ch. 5 - Prob. 98CRE Ch. 5 - Prob. 99CRE Ch. 5 - Prob. 100CRE Ch. 5 - Prob. 101CRE Ch. 5 - Prob. 102CRE Ch. 5 - Prob. 103CRE Ch. 5 - Prob. 104CRE Ch. 5 - Prob. 105CRE Ch. 5 - Prob. 106CRE Ch. 5 - Prob. 107CRE Ch. 5 - Prob. 108CRE Ch. 5 - Prob. 109CRE Ch. 5 - Prob. 110CRE Ch. 5 - Prob. 111CRE Ch. 5 - Prob. 112CRE Ch. 5 - Prob. 113CRE Ch. 5 - Prob. 114CRE Ch. 5 - Prob. 115CRE Ch. 5 - Prob. 116CRE Ch. 5 - Prob. 117CRE Ch. 5 - Prob. 118CRE Ch. 5 - Prob. 119CRE Ch. 5 - Prob. 120CRE Ch. 5 - Prob. 121CRE Ch. 5 - Prob. 122CRE Ch. 5 - Prob. 123CRE Ch. 5 - Prob. 124CRE Ch. 5 - Prob. 125CRE Ch. 5 - Prob. 126CRE

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