Finding the volume using cylindrical shells

[ThiagoG]

Homework Statement

x=1+(y-2)^², x=2. Rotating about the x-axis

Homework Equations

Volume=(2∏y)(1+(y-2)²(Δy)

Limits of integration would be from 1 to 3
2∏∫(y)(1+(y-2)²dy
2∏∫y³-4y²+5y dy

The Attempt at a Solution

2∏[y⁴/4-4y³/3+5y²/2]

Plug in the limits and I get 32∏/3. The answer is 16∏/3. It works if for the circumference you use 2∏ instead of 2∏y.

 

[eumyang]

Homework Statement

x=1+(y-2)^², x=2. Rotating about the x-axis

Homework Equations

Volume=(2∏y)(1+(y-2)²(Δy)

Limits of integration would be from 1 to 3
2∏∫(y)(1+(y-2)²)dy

Here's where you've gone wrong, I think. The representative rectangles need to be inside the parabola, not outside, as you have set the integral up.

 

[ThiagoG]

Here's where you've gone wrong, I think. The representative rectangles need to be inside the parabola, not outside, as you have set the integral up.

How would you fix this?

 

[eumyang]

(To help visualize what you did earlier, I attached two pics. The red region is what is being rotated around the x-axis. The "Wrong.bmp" file shows what you did, and the "Right.bmp" file shows what the problem is asking.)

The way you had set it up, your heights of the representative rectangle (or the distance from the y-axis to the parabola) is x (which equals 1 + (y - 2)²). Given that the distance from the y-axis to the line x=2 is 2, can you tell me the height of a rectangle from the parabola to the x=2 line?

 

[ThiagoG]

(To help visualize what you did earlier, I attached two pics. The red region is what is being rotated around the x-axis. The "Wrong.bmp" file shows what you did, and the "Right.bmp" file shows what the problem is asking.)

The way you had set it up, your heights of the representative rectangle (or the distance from the y-axis to the parabola) is x (which equals 1 + (y - 2)²). Given that the distance from the y-axis to the line x=2 is 2, can you tell me the height of a rectangle from the parabola to the x=2 line?

So it would be 3+(y-2)²?

edit: I tried this and it didn't work

 

[eumyang]

No, you don't want to add 2 and 1 + (y - 2)².

 

[ThiagoG]

No, you don't want to add 2 and 1 + (y - 2)².

I subtracted 2 from it. So my integral is 2∏∫(y)(-1+(y-1)²)dy. I got a negative number when I did this.

 

[eumyang]

Almost. Switch the order of subtraction.

 

[ThiagoG]

Almost. Switch the order of subtraction.

Ok thank you so much for your help. :)