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 said: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
eumyang said: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.
eumyang said:(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?
eumyang said:No, you don't want to add 2 and 1 + (y - 2)2.
eumyang said:Almost. Switch the order of subtraction.
The formula for finding the volume using cylindrical shells is V = ∫2πrh dx, where r is the radius of the shell, h is the height of the shell, and dx is the thickness of the shell.
The volume of a solid can be calculated using cylindrical shells by breaking the solid into thin cylindrical shells and adding up the volumes of each shell using the formula V = ∫2πrh dx.
Cylindrical shells is a method for finding volume that involves integrating the surface area of a cylinder. Other methods, such as the disk method or the washer method, involve integrating the cross-sectional area of a solid.
No, the cylindrical shells method can only be used for solids that have a cylindrical shape or can be broken down into cylindrical shells.
One limitation of using the cylindrical shells method is that it can be more difficult to set up the integral than other methods. Additionally, it may not be the most efficient method for finding volume in some cases.