Find the volume of the solid revolved around a region

[yesiammanu]

Homework Statement

Find the volume of the solid generated by revolving the region bounded by the graphs of y²=4x, the line y=x, about
A) x=4
B) y=4


So first I start out by graphing it

The intercepts are at 0,0 and 4,4

I use the washers method since there is a gap in between the line and the rotated solid, making a space in the middle of the solid

The washers method says V= ∏∫ ([R(x)]² - [r(x)²) where R(x) is the largest(outer) area, and r(x) is the smallest(inside) area. This is for rotating about the x-axis but can be used to rotate around the y axis. However, this isn't rotating about either of these axis; rather, it's rotating around x=4 which is what I am having trouble with.

So boundaries are 0 to 4, equation is ∏∫([y²/4]² - y²)dy but this is wrong. How would I do it correctly? Is cylindrical shells a better method? If I was using cylindrical shells, would it be 2∏∫(2√x - x)(x) since I use ∫ 2∏(shell height)(shell radius) Also I'm a bit confused on how I would do part B

Any help would be great, thanks.

 

[yesiammanu]

For part B, also using the cylindrical shells method, would the proper integral be 2∏∫(y - y²/4)(y)

 

[Dick]

Homework Statement

Find the volume of the solid generated by revolving the region bounded by the graphs of y²=4x, the line y=x, about
A) x=4
B) y=4


So first I start out by graphing it

The intercepts are at 0,0 and 4,4

I use the washers method since there is a gap in between the line and the rotated solid, making a space in the middle of the solid

The washers method says V= ∏∫ ([R(x)]² - [r(x)²) where R(x) is the largest(outer) area, and r(x) is the smallest(inside) area. This is for rotating about the x-axis but can be used to rotate around the y axis. However, this isn't rotating about either of these axis; rather, it's rotating around x=4 which is what I am having trouble with.

So boundaries are 0 to 4, equation is ∏∫([y²/4]² - y²)dy but this is wrong. How would I do it correctly? Is cylindrical shells a better method? If I was using cylindrical shells, would it be 2∏∫(2√x - x)(x) since I use ∫ 2∏(shell height)(shell radius) Also I'm a bit confused on how I would do part B

Any help would be great, thanks.

Nice picture. For the first one, if you are rotating around x=4, then wouldn't the inner radius be 4-y and the outer radius be 4-y^2/4? The radii should be distances to the axis of rotation.

 

[yesiammanu]

Ah I see, that makes sense, thank you. Is what I posted above (2∏∫(2√x - x)(x) for x=4, 2∏∫(y - y2/4)(y) for y=4) correct for cylindrical shell method?

 

[Dick]

Ah I see, that makes sense, thank you. Is what I posted above (2∏∫(2√x - x)(x) for x=4, 2∏∫(y - y2/4)(y) for y=4) correct for cylindrical shell method?

Same problem. I think you have the lengths of the shells right, but you don't have the radius right. You aren't rotating around 0.