text: # Area of Surface of Revolution
  
  1. Aug 3, 2009
  
  ### TheLegace
  
  1. The problem statement, all variables and given/known data
  Find area of surface generated by revolving about x-axis.
  
  y=x^3/3 1<=x<=sqrt(7)
  
  2. Relevant equations
  find f'(x) = x^2
  
  3. The attempt at a solution
  
  A = integral[ (x^3/3) * [(1+(x^2)^2] ^(1/2) ] ]dx
  = integral[ (x^3/3) * [(1+x^4) ^(1/2) ]dx
  I just don't know where to go from here.
  
  Any help is appreciated,
  Thank You.
  
  2. Aug 3, 2009
  
  ### Staff: Mentor
  
  Ordinary substitution: let u = 1 + x^4.
  
  I'm assuming that you have the correct integrand. If so, this substitution will help you out.
  
  3. Aug 3, 2009
  
  ### nickmai123
  
  You forgot the $$2\pi$$ in your equation, but you probably have that on paper. Here's what the integral should look like.
  
  $$A=\frac{2\pi}{3}\int^{\sqrt{7}}_{1}{x^{3}\sqrt{1+x^{4}}dy$$
  
  4. Aug 3, 2009
  
  ### TheLegace
  
  Oh whoops, ok well having that in there, how could I continue this problem?
  
  EDIT: I see how this can be solved, substition method.
  
  Thank You.
  
  Last edited: Aug 3, 2009
  5. Aug 3, 2009
  
  ### nickmai123
  
  Do what Mark44 said to do.
  
  $$u=1+x^{4}$$
  
  $$du=4x^{3}dx$$
  
  Last edited: Aug 3, 2009
  6. Aug 3, 2009
  
  ### Staff: Mentor
  
  Make that du = 4x3dx
  
  7. Aug 3, 2009
  
  Haha oops.
explanation: The document is a discussion thread from a math forum where a user is seeking help with a problem involving the calculation of the surface area of a solid of revolution. The user presents the problem, their attempt at a solution, and asks for help on how to proceed. Other users provide feedback and suggestions, including the correct integral and a hint to use substitution. The document contains a question and an ongoing discussion that leads to a solution.
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: # Area of Surface of Revolution
  
  1. Aug 3, 2009
  
  ### TheLegace
  
  1. The problem statement, all variables and given/known data
  Find area of surface generated by revolving about x-axis.
  
  y=x^3/3 1<=x<=sqrt(7)
  
  2. Relevant equations
  find f'(x) = x^2
  
  3. The attempt at a solution
  
  A = integral[ (x^3/3) * [(1+(x^2)^2] ^(1/2) ] ]dx
  = integral[ (x^3/3) * [(1+x^4) ^(1/2) ]dx
  I just don't know where to go from here.
  
  Any help is appreciated,
  Thank You.
  
  2. Aug 3, 2009
  
  ### Staff: Mentor
  
  Ordinary substitution: let u = 1 + x^4.
  
  I'm assuming that you have the correct integrand. If so, this substitution will help you out.
  
  3. Aug 3, 2009
  
  ### nickmai123
  
  You forgot the $$2\pi$$ in your equation, but you probably have that on paper. Here's what the integral should look like.
  
  $$A=\frac{2\pi}{3}\int^{\sqrt{7}}_{1}{x^{3}\sqrt{1+x^{4}}dy$$
  
  4. Aug 3, 2009
  
  ### TheLegace
  
  Oh whoops, ok well having that in there, how could I continue this problem?
  
  EDIT: I see how this can be solved, substition method.
  
  Thank You.
  
  Last edited: Aug 3, 2009
  5. Aug 3, 2009
  
  ### nickmai123
  
  Do what Mark44 said to do.
  
  $$u=1+x^{4}$$
  
  $$du=4x^{3}dx$$
  
  Last edited: Aug 3, 2009
  6. Aug 3, 2009
  
  ### Staff: Mentor
  
  Make that du = 4x3dx
  
  7. Aug 3, 2009
  
  Haha oops.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread from a math forum where a user is seeking help with a problem involving the calculation of the surface area of a solid of revolution. The user presents the problem, their attempt at a solution, and asks for help on how to proceed. Other users provide feedback and suggestions, including the correct integral and a hint to use substitution. The document contains a question and an ongoing discussion that leads to a solution.
  
  Label: question-and-answer