Explanation Obtain the radius and height of the solid of rotation to calculate the volume, to prove the given statement. Given: The bell-shaped curve y = f ( x ) in Figure 16 satisfies d y d x = − x y Calculation: y = f ( x ) in Figure 16 is an exponential function. When the region R is rotated about the y -axis, each shell has radius x and height f ( x ) . The volume of the solid will then be V = 2 π ∫ 0 a x f ( x ) d x Assuming u = f ( x ) , d u = f ' ( x ) = − x f ( x ) d x ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 6.4, Problem 62E

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To show that the solid obtained by rotating the region $R$ about the $y$ -axis has volume $V = 2 π (1 - c)$ where $c = f (a)$ , by using the Shell Method and the substitution $u = f (x)$ . Also observe that as $c \to 0$ , the region $R$ becomes infinite but the volume approaches $2 π$

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Chapter 6.4, Problem 61E

Chapter 6.5, Problem 1PQ

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To show that the solid obtained by rotating the region R about the y -axis has volume V = 2 π ( 1 − c ) where c = f ( a ) , by using the Shell Method and the substitution u = f ( x ) . Also observe that as c → 0 , the region R becomes infinite but the volume approaches 2 π

Solution Summary: The author proves that the solid obtained by rotating the region R about the y -axis has volume V = 2 ( 1

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To show that the solid obtained by rotating the region $R$ about the $y$ -axis has volume $V = 2 π (1 - c)$ where $c = f (a)$ , by using the Shell Method and the substitution $u = f (x)$ . Also observe that as $c \to 0$ , the region $R$ becomes infinite but the volume approaches $2 π$

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Chapter 6 Solutions

To show that the solid obtained by rotating the region R about the y -axis has volume V = 2 π ( 1 − c ) where c = f ( a ) , by using the Shell Method and the substitution u = f ( x ) . Also observe that as c → 0 , the region R becomes infinite but the volume approaches 2 π

Solution Summary: The author proves that the solid obtained by rotating the region R about the y -axis has volume V = 2 ( 1

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To show that the solid obtained by rotating the region R about the y-axis has volume V=2π(1−c) where c=f(a), by using the Shell Method and the substitution u=f(x). Also observe that as c→0, the region R becomes infinite but the volume approaches 2π

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Chapter 6 Solutions

To show that the solid obtained by rotating the region $R$ about the $y$ -axis has volume $V = 2 π (1 - c)$ where $c = f (a)$ , by using the Shell Method and the substitution $u = f (x)$ . Also observe that as $c \to 0$ , the region $R$ becomes infinite but the volume approaches $2 π$