Successive Integrals: Examining 1st Integral y_1(t)

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In summary, the conversation is discussing a textbook example of successive integrals and questioning if the first integral should be taken with regards to Tau instead of t. The person believes it is a typo and the integral should be taken with regards to Tau. The other person agrees and thanks them for catching the mistake.
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SuperCat
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My textbook shows the following example of successive integrals:

Does the 1st integral y_1(t) have a typo? Shouldn't the integral be taken with regards to Tau?
If not, I would appreciate an explanation of why the 1st successive integral is taken with regards to t, but in the second it is taken with regards to Tau.
 
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SuperCat said:
My textbook shows the following example of successive integrals:

bea3edbefa003bb45e863894e4a9b56f.png


Does the 1st integral y_1(t) have a typo? Shouldn't the integral be taken with regards to Tau?
If not, I would appreciate an explanation of why the 1st successive integral is taken with regards to t, but in the second it is taken with regards to Tau.
I believe you're right about this being a typo.
The integral should be ##\int_{-\infty}^t y(\tau) d\tau##.
 
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Mark thank you for your reply. I had a feeling that it didn't make sense, but I always feel a lot more comfortable getting a second opinion. This is now resolved.
 

FAQ: Successive Integrals: Examining 1st Integral y_1(t)

1. What is the purpose of examining the 1st integral in successive integrals?

The purpose of examining the 1st integral in successive integrals is to find the area under the curve of the function y_1(t). This can help in solving various problems in mathematics and physics, such as finding the displacement or velocity of an object.

2. How is the 1st integral related to the original function?

The 1st integral is the anti-derivative of the original function. This means that the derivative of the 1st integral will give the original function. In other words, the 1st integral is the reverse process of differentiation.

3. What is the significance of successive integrals?

Successive integrals allow us to find the area under a curve with more complex shapes by breaking it down into smaller and simpler parts. This is useful in solving real-world problems, as well as in theoretical mathematics.

4. How do you solve for the 1st integral?

To solve for the 1st integral, you need to find the anti-derivative of the original function y_1(t). This can be done by using integration techniques such as u-substitution, integration by parts, or partial fractions.

5. Can the 1st integral be used to solve definite integrals?

Yes, the 1st integral can be used to solve definite integrals. By finding the anti-derivative of the original function, we can then use the Fundamental Theorem of Calculus to evaluate the definite integral between two given limits.

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