25. fď S7 2e²dy dz
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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13.2 Double
Transcribed Image Text:**17–26. Evaluating Integrals**
Evaluate the following integrals as they are written.
![**25. Evaluate the double integral:**
\[
\int_0^1 \int_0^x 2e^{x^2} \, dy \, dx
\]
**Explanation:**
This is a double integral where the function \(2e^{x^2}\) is integrated first with respect to \(y\) and then with respect to \(x\).
- The limits of the inner integral are from 0 to \(x\), indicating that for each fixed \(x\), \(y\) varies between 0 and \(x\).
- The outer integral varies \(x\) from 0 to 1.
In the context of double integrals, this setup suggests calculating the area under a surface \(z = 2e^{x^2}\) over the triangular region defined by \(0 \leq y \leq x \leq 1\). The evaluation involves first integrating the function over \(y\), and then using that result to integrate over \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09613d8c-4cef-4639-a65c-ee2add319cba%2F98e85b07-a33d-47ef-9a79-060120aef8b0%2Flhikpic_processed.png&w=3840&q=75)
Transcribed Image Text:**25. Evaluate the double integral:**
\[
\int_0^1 \int_0^x 2e^{x^2} \, dy \, dx
\]
**Explanation:**
This is a double integral where the function \(2e^{x^2}\) is integrated first with respect to \(y\) and then with respect to \(x\).
- The limits of the inner integral are from 0 to \(x\), indicating that for each fixed \(x\), \(y\) varies between 0 and \(x\).
- The outer integral varies \(x\) from 0 to 1.
In the context of double integrals, this setup suggests calculating the area under a surface \(z = 2e^{x^2}\) over the triangular region defined by \(0 \leq y \leq x \leq 1\). The evaluation involves first integrating the function over \(y\), and then using that result to integrate over \(x\).
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