Are the following two integrals equivalent? [basic]

In summary, the conversation discusses the equivalence between two images and the acceptability of substituting between them when dealing with the area between two curves. It is confirmed that the equality is true and there are no exceptions, making it acceptable to use on an exam.
  • #1
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2
Hi, this is rather short one. I'm wondering if the two in the image below are equivalent.
http://gyazo.com/62449de717e91b412f42771fd7d0a1af

To me they appear to be equivalent, and I can't recognize any exceptions or trivial cases. Also, if it is equivalent would be it be acceptable to substitute between the two?

Specifically when dealing with the area between two curves. I know it is simple, and I know how, to identify the 'top' function but I'm always looking for new short cuts.

Thank you!
 
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  • #2
Yes, the equality is true.
 
  • #3
Alright thank you. I will assume since there are no exceptions, that is it fine to use on an exam or something similar.

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1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total value of a function over a specific interval.

2. What does it mean for two integrals to be equivalent?

When two integrals have the same value, they are considered equivalent. This means that they represent the same area under the same curve.

3. How can you tell if two integrals are equivalent?

To determine if two integrals are equivalent, you can use the Fundamental Theorem of Calculus, which states that if two integrals have the same integrand and the same limits of integration, then they are equivalent.

4. Can two integrals with different integrands be equivalent?

No, two integrals with different integrands cannot be equivalent. The integrand, which is the function being integrated, determines the value of the integral. Therefore, different integrands will result in different integral values.

5. Why is it important to know if two integrals are equivalent?

Knowing if two integrals are equivalent allows us to simplify complex integrals by replacing them with equivalent integrals that are easier to solve. It also helps us to check our work and ensure that our calculations are correct.

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