Integration By Volume

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1. Find the volume for the solid obtained by rotating the region bounded by the given curves about the specified line. y=(1/4)(x^2), x = 2, y = 0, about the y axis



2. Use the disk/washer method of integration to find the volume



3. When I try to solve this problem using small changes in x (dx) I get the wrong answer. Method: Take integral from 0 to 2 of a function for the area of a circle.

∫ from 0 to 2 : 2[(∏)*((x^2)/8)^2] dx → ∫ from 0 to 2 : (∏x^4)/32 → ∏/5

I do not see why I need to use the washer method as using the disk method encounters no hollow space

the answer given is 2pi

Please tell what I'm doing wrong.

Thanks

I'm not comfortable with the washer method, that's why I solved the problem with small change in the x axis.
 

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  • #2
Dick
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You appear to be rotating around the x-axis. The problem asks you to rotate around the y-axis. Then there is a hollow space between x=0 and x=sqrt(4y). The part between x=sqrt(4y) and x=2 is the filled in part.
 
  • #3
eumyang
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1. Find the volume for the solid obtained by rotating the region bounded by the given curves about the specified line. y=(1/4)(x^2), x = 2, y = 0, about the y axis



2. Use the disk/washer method of integration to find the volume



3. When I try to solve this problem using small changes in x (dx) I get the wrong answer. Method: Take integral from 0 to 2 of a function for the area of a circle.

∫ from 0 to 2 : 2[(∏)*((x^2)/8)^2] dx → ∫ from 0 to 2 : (∏x^4)/32 → ∏/5

If you're supposed to rotate around the y-axis and use the disk/washer method, then you need to integrate with respect to y, not x. Rewrite [itex]y = \frac{1}{4}x^2[/itex] in terms of x. Use x = 2 to find the upper limit of integration in terms of y. Also, this integral should be multiplied by π, not 2π.


EDIT: Beaten to it. ;)
 
  • #4
Dick
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EDIT: Beaten to it. ;)
Oh, not by much. Both posts have valuable information.
 
  • #5
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Thanks for all your help

I am rotating about the y axis

I am multiplying the entire ∏r2 portion by 2 because my equation only finds the area one half of the resultant solid

to find r I divided (1/4x2) by 2 to find the radius of my circles.

thanks again
 

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  • #6
eumyang
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I am multiplying the entire ∏r2 portion by 2 because my equation only finds the area one half of the resultant solid
No, you don't need to do that! The problem, as you stated it, says to
1. Find the volume for the solid obtained by rotating the region bounded by the given curves about the specified line. y=(1/4)(x^2), x = 2, y = 0, about the y axis
The resulting area of the region lies only in the 1st quadrant, and that is the only region that is revolving around the y-axis. There is no need to "multiply by 2."

to find r I divided (1/4x2) by 2 to find the radius of my circles.
I don't understand what you are saying here. You need to use the washer method, so you have to find two radii, normally notated as R and r. The integrand would contain R2 - r2, and pi is outside of the integral.
 

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