Volume of a Solid by Revolution

In summary, the problem is to find the volume of a solid obtained by rotating the region bounded by y=8*x+32, y=0 about the y-axis.
  • #1
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Homework Statement


Hi, the next problem I thought it was easy...and I really think it is, but I haven't come with the right answer :S. I compare the answer in an internet page of problems and all it says is "wrong"...

The problem is the following:
Find the volume of the solid obtained by rotating the region bounded by
y=8*x+32, y=0 about the y-axis.

Homework Equations


I made the sketh in Maple:
http://img109.imageshack.us/img109/5669/solidvolumexi5.th.jpg

The Attempt at a Solution


Then I solve for x:

x=(y/8)-4
and wrote the Integral with respect to y, and limits from 0 to 32 (where the function y=8x+32 intersects the y-axis).
Integral of Pi*f(x)^2, from y=0 to y=32, it should give me the volume, isn't it?

http://img164.imageshack.us/img164/98/equationie6.th.jpg
My answer: 512/3 * Pi= 536.1651462

But the internet page says I'm wrong but I don't know why :S

Any help is welcome, Thanks in advance!
 
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  • #2
Your answer looks right. I'd say the super trustworthy internet is wrong ;). (Disclaimer: I'm getting back into calc after a year of none, but still I'm quite sure you're correct).
 
  • #3
Possibly: Integration of Pi * (8x + 32)^2. -4 (lower limit) and 0 (upper limit)
 
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  • #4
Thanks to all of you for your answers.

AngeloG, I already tried what you suggest and I came with the answer:
4289.321169 which is wrong as well.

And because the region should be rotated about the Y-Axis and not about the X-axis, that's why I solved the equation for X, in order to have only Y's so I could integrate with respect to dy, then my limits are Y=0 and Y=32.

I hope I didn't understand you another thing, maybe "4289.321169" wasn't what you get, so I would like to know your answer.

Thx again^^, cheers!
 
  • #5
Err, forgot to say change the x's for y's and the y's for x's.

8x + 32 is basically the same as 8y + 32. One is just tilted on it's side, which is the y. Then you can integrate from -4 to 0, then rotate it around y-axis tilted.

(4096 / 3) * Pi, which is ~4289.3

However, I am wrong =). I was just doing the other half, considering your half was wrong. There's only two ways to do this and if both provide the wrong answer. Might be the page.
 
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  • #6
y= 8x+ 32 is a straight line with intercepts (0, 32) and (-4,0), rotating around the y-axis gives a cone with height 32 and radius 4. you can check the volume by using the standard formula for volume of a cone: [itex]V= \frac{1}{3}\pi r^2h[/itex].
 
  • #7
Thx again for all your answers^^, I didnt know there will be so many options or ways to solve the problem.

AngeloG, thanks for your reply.
HallsofIvy, I tried that :O, and I got: 536.1651462, but the page says its wrong -.-, so now I'm almost sure that is the webpage.

cheers!
 
  • #8
It might be a significant digits problem. Did you try just "536"?
 

1. What is meant by "Volume of a Solid by Revolution"?

The volume of a solid by revolution refers to the method of finding the volume of a three-dimensional shape by rotating a two-dimensional cross section around a specific axis.

2. How is the volume of a solid by revolution calculated?

The volume of a solid by revolution is calculated using the formula V = π∫ab(f(x))2dx, where π is the constant pi, a and b are the limits of integration, and f(x) is the function representing the shape's cross section.

3. What types of shapes can be measured using this method?

Any shape that can be formed by rotating a two-dimensional cross section around a line or axis can be measured using the volume of a solid by revolution method. This includes shapes such as cylinders, cones, spheres, and more complex shapes with irregular cross sections.

4. Are there any limitations to using this method?

One limitation of using the volume of a solid by revolution method is that it only applies to shapes that have a consistent cross section throughout their entire length. If the cross section changes at any point, a different method of finding the volume must be used.

5. How is the volume of a solid by revolution used in real-world applications?

The volume of a solid by revolution is used in various fields such as engineering, architecture, and physics to calculate the volume of objects or structures with rotational symmetry. This method is also commonly used in manufacturing and design processes for creating and measuring cylindrical or spherical objects.

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