Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by thegiven curves about the y-axis.y =y = 0,x = 0,X = 1Sketch the region and a typical shell.Step 1Rotating a vertical strip around the y-axis creates a cylinder with radius r = Xandheight h =4eSketch the region and a typical shell.4e-z²yy4-0.50.51lo1.51-1,5-1.0-0.50.51.01.5yy-0.50.51Jo1.5-1.5-1.01.01.5Step 2Now we can say that the volume of the solid created by rotating the region under y = 4eX and above the x-axis between x = 0 and x = 1 around the y-axis isV =2nrh dx1dx.4e4e-2Step 3The integral 2T4xe-x dx can be done with the substitution u =and du =-2xdx.-2.xStep 44xe-x dx = -4T4n- e"With the substitution, we have 2neu du = -474re"+C.Step 5Going back to x, the volume of our solid is4ле— 4ле

Answered: 2. 4л е -1 4ле 4ле

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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Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the
given curves about the y-axis.
y =
y = 0,
x = 0,
X = 1
Sketch the region and a typical shell.
Step 1
Rotating a vertical strip around the y-axis creates a cylinder with radius r = X
and
height h =
4e
Sketch the region and a typical shell.
4e-z²
y
y
4
-0.5
0.5
1lo
1.5
1
-1,5
-1.0
-0.5
0.5
1.0
1.5
y
y
-0.5
0.5
1Jo
1.5
-1.5
-1.0
1.0
1.5
Step 2
Now we can say that the volume of the solid created by rotating the region under y = 4eX and above the x-
axis between x = 0 and x = 1 around the y-axis is
V =
2nrh dx
1
dx.
4e
4e-2
Step 3
The integral 2T
4xe-x dx can be done with the substitution u =
and du =
-2x
dx.
-2.x
Step 4
4xe-x dx = -
4T
4n- e"
With the substitution, we have 2n
eu du = -
47
4re"
+C.
Step 5
Going back to x, the volume of our solid is
4ле
— 4ле

Expert Solution

Step 1

As you have solved all the parts of the question except the last part,so we will solve that part of the question which is unsolved that is the step 5.

We know when we have to calculate f(x) from a ( lower limit) to b (upper limit) we calculate f(b) and f(a) then take the difference that is we have to calculate

f(b)-f(a).

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