Volume of a Solid: Solve Problem with Washer/Shell Method

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The discussion focuses on solving a volume problem involving the rotation of a region bounded by the curves y=x^2, y=0, x=2, and x=4 about the line x=7. The user initially struggled with applying the washer and shell methods correctly, particularly in determining the inner and outer radii. After guidance, they recalculated using the shell method with the correct radius of 7-x and integration bounds from x=2 to x=4. The final volume obtained was confirmed as 424pi/3, resolving the homework challenge successfully. The discussion highlights the importance of correctly identifying radii and integration limits in volume calculations.
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I'm have trouble with a homework problem having to do with find volume of an area. the problem reads:

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

y=x^2 , y=0 , x=2 , x=4;


about the line x=7

I've tried using the washer method and also the shell method with no luck. Any help would be greatly appreciated.

Thanks
 
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Show your attempt, please, so as we can find out what went wrong.

ehild
 
since its rotating about the line x=7, I solved for y first

Im using pi*intergral from 0 to 16 of ((7-sqrt(y))^2-(7-5)^2)dy

I think I'm messing up with what my inner and outer radius' are and possibly the bounds
 
Last edited:
Y goes from 0 to x^2. First I would get the volume of a cylindrical shell of radius 7-x, thickness dx and height y=x^2. Then the integration with respect to x goes from x=2 to x=4.

ehild
 
Last edited:
That did it!
I came up with 424pi/3

When I initially tried the shell method I was using just x for the radius instead of 7-x. Thanks again for the help!
 
Is not it 436pi/3?

ehild
 
I double checked it and still got 424pi/3
The homework is online and I got it correct
 

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