BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 6.3, Problem 34E
To determine

To express: The limit (a) in terms of volume of solid (V) using the method of cylindrical shell.

Expert Solution

Answer to Problem 34E

The expression of the limit (a) in terms of V is 0.159V+0.6667_.

Explanation of Solution

Given:

The co-ordinates of the triangular region (x,y) are (0,0),(1,0), and (1,2).

The equations are y=2x, x=a; where a>1.

Calculation:

Plot a graph using the co-ordinates of the triangular region (x,y) are (0,0),(1,0), and (1,2):

Draw the shell as shown in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 6.3, Problem 34E

Calculate the volume using the method of cylindrical shell:

V=ab2πx[f(x)]dx (1)

Substitute 0 for a, 1 for b, (ax) for x, and 2x for [f(x)] in Equation (1).

V=012π(ax)(2x)dx=4π01(axx2)dx (2)

Integrate Equation (2).

V=4π[a(x1+11+1)(x2+12+1)]01=4π[a2x213x3]01=4π[(a×122133)0]

=4π(0.5a13)V=6.283a4.18879 (3)

Rearrange Equation (3).

6.283a=V+4.18879a=V+4.188796.283=0.159V+0.6667

Hence, the expression of the limit (a) in terms of V is 0.159V+0.6667_.

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