Can I Trust Blind Substitutions in Integration?

In summary, when it comes to finding the area under a curve in calculus, it is important to consider whether you are interested in the net area or the absolute area. If you are interested in the net area, the curve passing beyond the negative side does not matter. However, for the absolute area, the integral may need to be split into positive and negative parts. It is also important to be careful with substitutions and consider the limits of integration to avoid making mistakes.
  • #1
dilan
72
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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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 

FAQ: Can I Trust Blind Substitutions in Integration?

What is integration and why is it important in science?

Integration is the process of combining different elements or parts into a whole. In science, it is important because it helps us understand how different parts of a system or phenomenon work together to create a larger picture.

What are some common challenges with integrating different scientific theories or concepts?

One of the main challenges with integrating scientific theories or concepts is finding a way to reconcile conflicting evidence or ideas. It can also be difficult to bridge the gap between different disciplines or fields of study.

How can integration be used to solve problems in science?

Integration can be used to solve problems in science by providing a more comprehensive understanding of a topic or phenomenon. It allows scientists to combine different perspectives and approaches to find solutions to complex problems.

What are some strategies for effectively integrating different scientific ideas or theories?

Some strategies for effectively integrating different scientific ideas or theories include identifying common themes or patterns, utilizing interdisciplinary approaches, and constantly reevaluating and adjusting the integration process.

Can integration ever be detrimental to scientific progress?

While integration can be a useful tool in science, it can also be detrimental if it is forced or used to dismiss conflicting evidence. It is important to approach integration with an open mind and continuously evaluate the validity and relevance of integrated ideas.

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