text: # Disk/Washer Method - check my set up?
  
  ## Homework Statement
  
  choose between washer and disk method, and find the area of the region bounded by the following curves by revolving around a) the y-axis b) the x-axis c) y=8, and x=2
  
  y=2x2
  y=0
  x=2
  
  ## The Attempt at a Solution
  
  So I set up a) from 0 to 8 [ 2-sqrt(y/2)]^2 dy (washer method)
  
  I don't know how to do the notation around here, so I hope that is clear? And I did remember the PI out front in these. I got 16pi/3
  
  b) from 0 to 2 (2x^2)^2 dx (disk method). for 128pi/5
  
  c) from 0 to 2 (x-2x^2)^2 dx (washer method) for 184pi/15
  
  d) from 0 to 8 (sqrt(y/2))^^2 dy (disk method) for 16pi
  
  I have no clue if I'm doing this right.... all the values seem so different... :(
  
  HallsofIvy
  Yes, for rotation around the y-axis, since the y-axis is not a boundary of the region being rotated, use the "washer method". However, you have the integrand wrong. $\pi (r_1- r_2)^2$ is the area of a full circle of radius $r_1- r_2$. A "washer" with outer radius $r_1$ and inner radius $r_2$ can be thought of as the are of the outer circle, $\pi r_1^2$ and then subtract of the area of the inner circle, $\pi r_2^2$: the area of the washer is $\pi(r_1^2- r_2^2)$.
  The radius will be along the x- direction and, since $y= 2x^2$ but we only need x positive, $x= y^{1/2}/\sqrt{2}$ and the area of the "washer" from that x to x= 2 is $\pi(r_1^2- r_2^2)= \pi(4- y/2)$. The volume is $\pi \int_0^8 (4- y/2)dy$.
explanation: The text extract is a discussion from a homework help forum where a user is seeking verification for their setup of problems involving the disk and washer methods for finding volumes of solids of revolution. The user provides their attempts at solving the problems and asks for confirmation if their setups and answers are correct. Another user, HallsofIvy, responds by correcting the user's integrand for part (a) and providing the correct setup.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Disk/Washer Method - check my set up?
  
  ## Homework Statement
  
  choose between washer and disk method, and find the area of the region bounded by the following curves by revolving around a) the y-axis b) the x-axis c) y=8, and x=2
  
  y=2x2
  y=0
  x=2
  
  ## The Attempt at a Solution
  
  So I set up a) from 0 to 8 [ 2-sqrt(y/2)]^2 dy (washer method)
  
  I don't know how to do the notation around here, so I hope that is clear? And I did remember the PI out front in these. I got 16pi/3
  
  b) from 0 to 2 (2x^2)^2 dx (disk method). for 128pi/5
  
  c) from 0 to 2 (x-2x^2)^2 dx (washer method) for 184pi/15
  
  d) from 0 to 8 (sqrt(y/2))^^2 dy (disk method) for 16pi
  
  I have no clue if I'm doing this right.... all the values seem so different... :(
  
  HallsofIvy
  Yes, for rotation around the y-axis, since the y-axis is not a boundary of the region being rotated, use the "washer method". However, you have the integrand wrong. $\pi (r_1- r_2)^2$ is the area of a full circle of radius $r_1- r_2$. A "washer" with outer radius $r_1$ and inner radius $r_2$ can be thought of as the are of the outer circle, $\pi r_1^2$ and then subtract of the area of the inner circle, $\pi r_2^2$: the area of the washer is $\pi(r_1^2- r_2^2)$.
  The radius will be along the x- direction and, since $y= 2x^2$ but we only need x positive, $x= y^{1/2}/\sqrt{2}$ and the area of the "washer" from that x to x= 2 is $\pi(r_1^2- r_2^2)= \pi(4- y/2)$. The volume is $\pi \int_0^8 (4- y/2)dy$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a discussion from a homework help forum where a user is seeking verification for their setup of problems involving the disk and washer methods for finding volumes of solids of revolution. The user provides their attempts at solving the problems and asks for confirmation if their setups and answers are correct. Another user, HallsofIvy, responds by correcting the user's integrand for part (a) and providing the correct setup.
  
  Label: question-and-answer