calculus help

[lilkrazyrae]

Find the volume of the solid generated by revolving the region about the given line.

the region in the first quadrant bounded above by the line y=4, below by the line y=4x/3, and on the left by the y axis, about the line y=4.

Please help!

 

[Guest]

find the intercepts for the graphs to establish the x coordinates....
then use intergration methods.
don you need more?

 

[Gene]

The volume of a ring-shell h high and r wide is
dV=2*pi*r*h*dx =
2*pi*(4-x)*(4-4x/3)*dx
You want the integral from x=0 to 3.

 

[lilkrazyrae]

help

I used the equation V=2pi*r*h and integrated it but I came up with an answer that was not correct the answer should be 16pi and I got 32pi. Please help me.

 

[Gene]

Got a problem with you and the book. I get
2*pi*r*h =
2*pi*(4-x)*(4-4x/3) =
2*pi*(16-4x-16x/3+4x²/3 =
2*pi*(16-(28/3)x + 4x²/3) =
8/3*pi(12-7x+x²)
integral(8/3*pi(12-7x+x²)dx =
8/3*pi(12x-7/2x²+x³/3)
Evaluate from x=0 to 3 that is
8/3*pi*(36-63/2+9)-0 =
36 pi

As a check:
Consider a cone of radius 4 and a height of 16/3
V=(1/3)bh =
(1/3)pi*r²*h =
(1/3)pi*4²*16/3 =
28&4/9 pi

Consider a cone of radius 1 and a height of (16/3)-4
V=(1/3)*pi*1²*4/3 =
=4/9 pi

The volume of the large cone below y=4 is the difference.
pi*(28&4/9-4/9)=28 pi

The volume of the disk of radius 4 and height 4 is
bh = pi*r²h =
64 pi

Your volume is the difference
(64-28) pi =
36 pi

If I didn't goof somewhere... Anyone????

 

[tkhunny]

You should be able to do this sort of thing both ways. It is an excellent check on your setup.

From the volume of a Right Circular Cylinder, pi*r²*h, differentiate with respect to each variable.

(d/dr)(pi*r²*h) = 2*pi*r*h
(d/dh)(pi*r²*h) = 2*pi*r²

This should give you a hint how to set up each version:

2*pi*r*h ==> Set up integration over the changing radius.
2*pi*r² ==> Set up integration over the changing height.

Integrate[0,4](2*pi*(4-y)*x) dy = Integrate[0,4](2*pi*(4-y)*(3/4)*y) dy = 16*pi

Integrate[0,3](pi*(4-y)²) dx = Integrate[0,3](pi*(4-(4/3)*x)²) dx = 16*pi

 

[Gene]

Arrrrgh :roll:
I rotated around x = 4, not y=4. If only I had learned to read.
Never mind.
------------------
Gene