Volume of solid of revolution question

[UCdavisEcon]

Revolve the region bounded by y=x+2, y=x^2, about the x-axis

I'm going to use the shell method, so i understand this will be in terms of y, so x=y-2, x=sqrt(y). I understand the formula is, V=2pi *integral from a to b of (radius of shell)(height of shell)dy

What i'm getting confused on is the radius and height of the shell. What i have down so far is V=2pi integral from 0 to 4 of (y)(y-2-sqrt(y). I see the radius as being just y, the height as being (y-2)-(sqrt(y)). The answer is my book is 72pi/5. Please help. thank you.

 

[Deleted member 4993]

Revolve the region bounded by y=x+2, y=x^2, about the x-axis

I'm going to use the shell method, so i understand this will be in terms of y, so x=y-2, x=sqrt(y). I understand the formula is, V=2pi *integral from a to b of (radius of shell)(height of shell)dy

I would have done it by disk method - but it can be done other way.

What i'm getting confused on is the radius and height of the shell. What i have down so far is V=2pi integral from 0 to 4 of (y)(y-2-sqrt(y). I see the radius as being just y, the height as being (y-2)-(sqrt(y)). The answer is my book is 72pi/5. Please help. thank you.

so you have:

\(\displaystyle \displaystyle{2 * \pi *\int_0^4 y * (y-2) * \sqrt{y} dy}\)

so what is the hangup....

 

[UCdavisEcon]

Well, that's not right. I see that the disk/washer method is far easier, however my homework tells me to use the shell method for practice. I still am not seeing how to set up the integral. what i set up was wrong. I don't understand how to represent the radius and height of the shells.

 

[Ishuda]

Revolve the region bounded by y=x+2, y=x^2, about the x-axis

I'm going to use the shell method, so i understand this will be in terms of y, so x=y-2, x=sqrt(y). I understand the formula is, V=2pi *integral from a to b of (radius of shell)(height of shell)dy

What i'm getting confused on is the radius and height of the shell. What i have down so far is V=2pi integral from 0 to 4 of (y)(y-2-sqrt(y). I see the radius as being just y, the height as being (y-2)-(sqrt(y)). The answer is my book is 72pi/5. Please help. thank you.

Why only 0 and 4? The two curves intersect at (2, 4) and (-1, 1) so the region would be -1\(\displaystyle \le\)x\(\displaystyle \le\)2 or the two intervals 0\(\displaystyle \le\)y\(\displaystyle \le\)1 and 0\(\displaystyle \le\)y\(\displaystyle \le\)4.