Equality of integrals VS equality of integrands

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  • Thread starter Ahmed Mehedi
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In summary, it can be proven using calculus of variation that if two continuous functions are different at a single point, then they differ on an interval.
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@Adesh I generally had very good professors in undergrad and learned a lot from their lectures. I can sometimes be a slow student though, so I generally had to re-work the parts I didn't follow afterwards, but this is all on me and certainly doesn't reflect negatively on my instructors.

I think Rudin is harder to self-study because it's terse and doesn't "hand-hold" the reader. This is much less of an obstacle when you have all the aspects of a class to help you along (other students, professors, teaching assistants, homeworks, etc.)

In the future, you might want to PM a user if you have a specific question for them, instead of posting on an unrelated thread.
 
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<h2>1. What is the difference between equality of integrals and equality of integrands?</h2><p>The equality of integrals means that two integrals have the same value, while equality of integrands means that the functions being integrated are equal. In other words, equality of integrals relates to the result, while equality of integrands relates to the process.</p><h2>2. Can two integrals with different integrands have the same value?</h2><p>Yes, it is possible for two integrals with different integrands to have the same value. This is because the value of an integral is dependent on the limits of integration and not just the integrand itself.</p><h2>3. How do you determine if two integrands are equal?</h2><p>To determine if two integrands are equal, you can use algebraic manipulation or substitution to simplify the expressions and see if they are identical. You can also graph the two functions to visually compare them.</p><h2>4. Is equality of integrands necessary for equality of integrals?</h2><p>No, equality of integrands is not necessary for equality of integrals. As long as the limits of integration and the function being integrated are the same, the integrals will have the same value.</p><h2>5. What are some applications of understanding the difference between equality of integrals and equality of integrands?</h2><p>Understanding the difference between equality of integrals and equality of integrands is important in calculus and other areas of mathematics. It can also be useful in physics and engineering, where integrals are used to solve problems related to motion, work, and other physical quantities.</p>

FAQ: Equality of integrals VS equality of integrands

1. What is the difference between equality of integrals and equality of integrands?

The equality of integrals means that two integrals have the same value, while equality of integrands means that the functions being integrated are equal. In other words, equality of integrals relates to the result, while equality of integrands relates to the process.

2. Can two integrals with different integrands have the same value?

Yes, it is possible for two integrals with different integrands to have the same value. This is because the value of an integral is dependent on the limits of integration and not just the integrand itself.

3. How do you determine if two integrands are equal?

To determine if two integrands are equal, you can use algebraic manipulation or substitution to simplify the expressions and see if they are identical. You can also graph the two functions to visually compare them.

4. Is equality of integrands necessary for equality of integrals?

No, equality of integrands is not necessary for equality of integrals. As long as the limits of integration and the function being integrated are the same, the integrals will have the same value.

5. What are some applications of understanding the difference between equality of integrals and equality of integrands?

Understanding the difference between equality of integrals and equality of integrands is important in calculus and other areas of mathematics. It can also be useful in physics and engineering, where integrals are used to solve problems related to motion, work, and other physical quantities.

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