Advanced Volume of Irregular object

mistyblinx

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Find the volume of a solid generated by revolving about the line y=-1 the region bounded by 2y=x+3 and the outside of the curves y^2+x=0 and y^2-4x=0.

I graphed it using Desmos to visualize what I needed to do. I think I'm looking for the 2 sections bounded by 2y=x+3 and y=-1, squeezed between the two parabolas. The range between (-1, 2) to be more specific. At this point I have tried to wrap my brain around what would be the be way to approach this question. Should I use the area formula for the 4 separate quadrants? For example in quadrant one take the integral of (2y=x+3)-(2y=x+3), (0,2) with respect to x and then do that for each section. They put the Area into the volume formula. Should I take the area from (-1,2) and then subtract the area of the parabolas? I think I'm just at a loss for where to start this problem. I would appreciate any help.
 

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You should apply the natural formula to every region necessary. Let's see your work.

Keep in mind that one integration direction, either dy or dx, may be simpler than the other. Personally, I like to do it both ways. This solidifies my result because I keep trying until I get the same answer both ways. It also gives me plenty of extra practice.
 
Find the volume of a solid generated by revolving about the line y=-1 the region bounded by 2y=x+3 and the outside of the curves y^2+x=0 and y^2-4x=0.

… I think I'm looking for the 2 sections bounded by 2y=x+3 and y=-1, squeezed between the two parabolas.
Note that the instruction says "region" (singular), not "regions". There's only one region to rotate; it lies within Quadrants I and II.


Should I use the area formula for the 4 separate quadrants?
No. You need to integrate rotated "slices" (rectangles) of the region.


They put the Area into the volume formula.
Who did this? What volume formula?

Here's a video showing a standard method:

https://www.khanacademy.org/math/ap...thod/v/washer-method-rotating-around-non-axis
 
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