Find the volume of the solid generated by revolving the standed region about the

totalmente80sm9 2021-11-21 Answered
Find the volume of the solid generated by revolving the standed region about the x-axis.

Te volume of the solid is ? cubic units.
The equation: 4x+3y=24
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Expert Answer

Philip O'Neill
Answered 2021-11-22 Author has 8 answers
Step 1
y=8(43)x
y=4×(2(13)x)
for x intercept, y=0
0=4×(2(13)x)
(13)x=2
x=6
Volume generated by rotating the given region about x axis by washer method v=06π[4×(2(13)x)]2dx
v=0616π[222×2(13)x+((13)x)2]dx
v=0616π[4(43)x+(19)x2]dx
v=0616π[4x(43)(12)x2+(19)(13)x3]
v=0616π[4x(23)x2+(127)x3]
v=16π[4×6(23)62+(127)63]16π[4×0(23)02+(127)03]
v=16π[4×6(2×363)+(21627)]
v=16π[2424+8]
v=128π
Volume =128π=402
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Witheyesse47
Answered 2021-11-23 Author has 14 answers

Step 1
The disk formed when the skethched region revolves around the x-axis is shown below:
image

The given equation is, 4x+3y=24
3y=244x
y=843x
To get x-intercept, equate y=0
y=843x=0
8=43x
x=6
Here, the disk radius y=f(x)=843x
And also x varies from 0 to 6.
Step 2
Volume of the solid of revolution obtained the given region about the x-axis is,
V=πab[f(x)]2dx
=π06[843x]2dx
=π06[64+169x2643x]dx
=π[64x+649(x33)643(x22)]06
=π[384+128384]
=128π
=402.12
Therefore the required volume of the solid is,
=128π cubic units
=402.12 cubic units
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