The discussion focuses on calculating the volume of a solid of revolution formed by rotating the region bounded by the curve x = 1 + (y - 2)² and the line x = 2 about the x-axis. The correct volume formula is derived using the cylindrical shells method, resulting in the integral 2∏∫(y)(1 + (y - 2)²) dy, with limits of integration from 1 to 3. The final volume calculation yields 16∏/3, correcting the initial miscalculation of 32∏/3 by properly setting up the representative rectangles within the parabola.
PREREQUISITESStudents studying calculus, particularly those focusing on volume calculations and solid geometry, as well as educators seeking to clarify the cylindrical shells method.
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.