Show steps please x= 4y = x|(2, 8)y = 8y = Vx(4, 2)Figure 1Figure 21. Use the specified technique to setup a definite integral that when evaluated gives thevolume of the solid generated when the indicated region is rotated about the indicatedline. You need not evaluate the integral.a) Figure 1 about the x =4 using disks.b) Fig.2 about y=-2 using rings.c) Figure 1 about the y-axis using shells.d) Figure 2 about y=9 using shells.

Answered: x = 4 y = x' |(2, 8) y = 8 y = Vx (4,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Show steps please

x= 4
y = x
|(2, 8)
y = 8
y = Vx
(4, 2)
Figure 1
Figure 2
1. Use the specified technique to setup a definite integral that when evaluated gives the
volume of the solid generated when the indicated region is rotated about the indicated
line. You need not evaluate the integral.
a) Figure 1 about the x =4 using disks.
b) Fig.2 about y=-2 using rings.
c) Figure 1 about the y-axis using shells.
d) Figure 2 about y=9 using shells.

Expert Solution

Step 1

Solution for question a:

In figure 1 the region is bounded $x = y^{2}$ ,the $x -$ axis, where $0 \leq y \leq 2$

The formula to find the volume of the solid formed by rotating the region bounded by the curves $x = f (y)$ , $x = g (y)$ , where $f (y) \leq g (y)$ and $c \leq y \leq d$ is:

$V = π \int_{c}^{d} ({(o u t e r r a d i u s)}^{2} - {(i n n e r r a d i u s)}^{2}) d y$

Here outer radius is the distance between the line $x = 4$ and the outer curve and the inner radius is the distance between the inner curve and the line $x = 4$ .

Here the inner curve is the line $x = 4$ and the outer curve is $x = y^{2}$ .

The distance between the inner curve and the line $x = 4$ is:

$= 4 - 4 = 0$

The distance between the outer curve and the line $x = 4$ is:

$= 4 - y^{2}$

Therefore, the volume of solid of revolution is,

$V = π \int_{0}^{2} ({(4 - y^{2})}^{2} - {(0)}^{2}) dy$

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