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Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.x2 + 4y2 = 4  About y=2About x=2

Question

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

x2 + 4y2 = 4

 

 About y=2

About x=2

 
 
check_circleAnswer
Step 1

Find the volume of the solid.

The given curve is r 4y 4
24-4y
x=t/4-4y2
Find the volume of the solid about the line y 2
Sketch the graph of region bounded by the curve +4y-4 about the line y 2 as
shown below in Figure 1
y=2
(0.1)
(-2,0)
2.0)
|(0.-1)
Figure
When using shell method, take slices through the region parallel to the x-axis. A typical
slice at a point y, where -s ys, has height
about the line y 2. it sweeps out a elliptical shell with radius 2- y
V4
-4y and these slice is revolved
help_outline

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The given curve is r 4y 4 24-4y x=t/4-4y2 Find the volume of the solid about the line y 2 Sketch the graph of region bounded by the curve +4y-4 about the line y 2 as shown below in Figure 1 y=2 (0.1) (-2,0) 2.0) |(0.-1) Figure When using shell method, take slices through the region parallel to the x-axis. A typical slice at a point y, where -s ys, has height about the line y 2. it sweeps out a elliptical shell with radius 2- y V4 -4y and these slice is revolved

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Step 2

Calculate the volume of the solid about y=2.

V =[27Rhdy
-2(2-y)24-4p
- rj4-y d=2j »{A=y")
-4y dy
Evaluate the integral 4/1-y dy
-1
Substitute y sinuusin'
dy cosudu
41-ydy 4cosui-sin udu
=
-1
y-1u
2
= 4| cos udu
y 1u =
2
1 cos2u
du
2
sin 2u
2 u+
2
help_outline

Image Transcriptionclose

V =[27Rhdy -2(2-y)24-4p - rj4-y d=2j »{A=y") -4y dy Evaluate the integral 4/1-y dy -1 Substitute y sinuusin' dy cosudu 41-ydy 4cosui-sin udu = -1 y-1u 2 = 4| cos udu y 1u = 2 1 cos2u du 2 sin 2u 2 u+ 2

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Step 3

On further simpli...

sin 2
2
sin 2
2
2
2
2
2
= 27T
j:(=F)-0.
(-y")
Since y
is odd function, it can be concluded that
Thus, the volume integral becomes
V 4T(2T -0)
78.95684
help_outline

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sin 2 2 sin 2 2 2 2 2 2 = 27T j:(=F)-0. (-y") Since y is odd function, it can be concluded that Thus, the volume integral becomes V 4T(2T -0) 78.95684

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