How can I solve a volume integral question with a trig substitution?

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Discussion Overview

The discussion revolves around solving a volume integral question using trigonometric substitution. Participants explore methods for evaluating the integral ∫^{1}_{-1} 4(√(1-x^{2}))(x+1)dx, which is part of a larger triple integral. The final answer is known to be 2π, but participants express uncertainty about the appropriate techniques to use.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion over how to solve the integral and mentions attempts with integration by substitution and integration by parts.
  • Another participant asserts that the integral is not a volume integral and suggests using the substitution x=sin u.
  • A participant clarifies that the integral is part of a triple volume integral and questions whether a simple substitution suffices or if integration by parts is also necessary.
  • Another participant indicates that the suggested substitution does work but requests to see the progress made with it.
  • A later reply notes the presence of the √(1 - x²) factor as a reason for using trigonometric substitution, aligning with earlier recommendations.
  • A participant points out spelling errors related to the terms "integral," "integrate," and "integration."

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to solve the integral, with multiple approaches and some uncertainty about the necessity of integration by parts alongside substitution.

Contextual Notes

Some participants indicate that the integral is part of a larger context (a triple integral), which may affect the approach needed. There is also uncertainty about the effectiveness of different substitution methods.

ppy
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Hi,

I was attempting a volume integral question out of a book. I know what the final answer is and what integral i am supposed to work out but I do not know how I am supposed to solve it. I have tried different ways such as integration by substitution and integration by parts but I do not seem to be getting anywhere.

This is the ∫^{1}_{-1} 4(√(1-x^{2}))(x+1)dx The answer is 2pi.

any help would be great thanks.
 
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This is not a volume integral. Put x=sin u and proceed
 
The integral started out as a triple volume integral. The part mentioned is the final part of the triple integral. I have tried x=sinu and still got nowhere. Is it a simple substitution or do I have to integrate by parts as well ?
Thanks
 
hi ppy! :smile:
ppy said:
I have tried x=sinu and still got nowhere. Is it a simple substitution or do I have to integrate by parts as well ?

davidmoore63@y's :smile: substitution does work …

show us how far you got with it :wink:​
 
Thanks. I've got there now !
 
The presence of the ##\sqrt{1 - x^2}## factor suggests that a trig substitution is called for, and that's the direction that davidmoore and tiny-tim are recommending.

BTW, there are no such words in English as "intergral", "intergrate", or "intergration".
 

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