Now we can say that the volume of the solid created by rotating the region under y = 4e- and above the x-axis between x = 0 and x = 1 around the y-axis is V = 2nrh dx dx. de? The integral 2n 4xe-x dx can be done with the substitution u = and du = dx. With the substitution, we have 2n 4xe- dx = - e du =- + C. Going bac x, the volume of our solid is

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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Now we can say that the volume of the solid created by rotating the region under y =
and above the x-axis between x = 0 and x = 1 around the y-axis is
V =
2nrh dx
2nX
4e
dx.
The integral 27T
4xe-x dx can be done with the substitution u =
and du =
dx. With the substitution, we have 2n
4xe-x?
dx = -
eu du = -
+ C. Going back to
x, the volume of our solid is
Transcribed Image Text:Now we can say that the volume of the solid created by rotating the region under y = and above the x-axis between x = 0 and x = 1 around the y-axis is V = 2nrh dx 2nX 4e dx. The integral 27T 4xe-x dx can be done with the substitution u = and du = dx. With the substitution, we have 2n 4xe-x? dx = - eu du = - + C. Going back to x, the volume of our solid is
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 = 4e-x?
y = 0,
X = 0,
x = 1
Sketch the region and a typical shell. Rotating a vertical strip around the y-axis creates a cylinder with radius r =
and height h =
Sketch the region and a typical shell.
y
y
4
2
-0.5
0.5
10
1.5
-0,5
0.5
10
1.5
y
y
2
-1.5
-1.0
-0.5
0.5
1.0
1.5
-1.5
-1.0
8.5
1.0
1.5
Transcribed Image Text: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 = 4e-x? y = 0, X = 0, x = 1 Sketch the region and a typical shell. Rotating a vertical strip around the y-axis creates a cylinder with radius r = and height h = Sketch the region and a typical shell. y y 4 2 -0.5 0.5 10 1.5 -0,5 0.5 10 1.5 y y 2 -1.5 -1.0 -0.5 0.5 1.0 1.5 -1.5 -1.0 8.5 1.0 1.5
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