Can I Trust Blind Substitutions in Integration?

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Discussion Overview

The discussion revolves around the topic of integration in calculus, specifically addressing concerns about the use of blind substitutions when calculating areas under curves, particularly when the curves extend into negative values. Participants explore the implications of different approaches to integration and the importance of understanding the conditions under which substitutions are applied.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant expresses a need for resources to better understand integration, particularly in the context of negative areas.
  • Another participant suggests that for net area calculations, the sign of the curve does not matter, while for absolute area, one should consider both positive and negative parts separately.
  • A different viewpoint emphasizes that unless specified, one should integrate the function as is within the given limits, but also highlights the need to calculate absolute areas by separating positive and negative contributions.
  • Concerns are raised about the potential pitfalls of using substitutions without careful consideration of the integrand and the limits of integration, warning that this can lead to incorrect results.

Areas of Agreement / Disagreement

Participants exhibit a mix of perspectives, with some agreeing on the general approach to integration while others emphasize caution regarding substitutions. The discussion reflects both agreement on basic principles and disagreement on the implications of those principles in practice.

Contextual Notes

Participants note the importance of understanding the conditions under which substitutions are valid and the necessity of considering the signs of the integrand, but do not resolve the nuances of these issues.

Who May Find This Useful

This discussion may be useful for students new to calculus, particularly those grappling with integration techniques and the implications of negative areas in their calculations.

dilan
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I am really new to the calculus field. I have a small problem in integration when it comes to finding the area under a curve when the curve passes beyond the negative side.
Can anyone provide me some links where I can get some good knowledge about integration ?

Thanks :smile:
 
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What's the problem? If you are interested in the net area, then it doesn't matter that the curve is negative. If you are interested in the absolute area, then you break up in the integral into positive and negative parts, and take the absolute value of the negative parts.
 
Never worry during taking the area.Unless otherwise stated just integrate the function within the given limit.If it is stated to say the total absolute area,calculate for +ve and -ve sides separately.
 
Never worry?

You should always be carefull when it comes to absolute values or substitution methods. What if your substitution is not well defined within the limits of integration, or you have to split the integral into two because of the signs of the integrand?

It's a classic mistake of first year calc students to do substitutions blindly, without concerning what is happening between the domain of integration and the integrand, only to end up with wrong answers.
 

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