Consider the following integral. Sketch its region of integration in the xy-plane. dx dy In(x) (a) Which graph shows the region of integration in the xy- el plane? ? e*3 (b) Write the integral with the order of integration reversed: B D A В dæ dy In(x) with limits of integration dy dx In(x) A A = B = C = D = D
Consider the following integral. Sketch its region of integration in the xy-plane. dx dy In(x) (a) Which graph shows the region of integration in the xy- el plane? ? e*3 (b) Write the integral with the order of integration reversed: B D A В dæ dy In(x) with limits of integration dy dx In(x) A A = B = C = D = D
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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Transcribed Image Text:(c) Evaluate the integral.
*Note: The image appears to show a blank box with a pencil icon, indicating an area where users might input or annotate an integral for evaluation. No graphs or additional diagrams are present.*
![### Transcription:
**Consider the following integral. Sketch its region of integration in the xy-plane.**
\[
\int_{0}^{3} \int_{e^y}^{e^3} \frac{x}{\ln(x)} \, dx \, dy
\]
**(a)** Which graph shows the region of integration in the xy-plane?
[Dropdown Menu]
**(b)** Write the integral with the order of integration reversed:
\[
\int_{0}^{3} \int_{e^y}^{e^3} \frac{x}{\ln(x)} \, dx \, dy = \int_{A}^{B} \int_{C}^{D} \frac{x}{\ln(x)} \, dy \, dx
\]
with limits of integration
- A = [ ]
- B = [ ]
- C = [ ]
- D = [ ]
### Graphs Explanation:
- **Graph A**: A region bounded by the y-axis from 0 to 3, an arc from \((e^y, y)\) to \((e^3, 3)\), and lines along \(x = e^3\).
- **Graph B**: Similar to Graph A, but mirrored over the x-axis.
- **Graph C**: A region bounded by the x-axis from \(e^0 (=1)\) to \(e^3\), a line from \(x = e^3\) to \(3\), and an arc along \((x, \ln(x))\).
- **Graph D**: Similar to Graph C, but mirrored over the y-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9f391b6-5f04-4d98-9bc5-d57c88046352%2F0bdcda9a-3a3c-48b5-bfb3-f56d4b6c3728%2Fep33e7p_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription:
**Consider the following integral. Sketch its region of integration in the xy-plane.**
\[
\int_{0}^{3} \int_{e^y}^{e^3} \frac{x}{\ln(x)} \, dx \, dy
\]
**(a)** Which graph shows the region of integration in the xy-plane?
[Dropdown Menu]
**(b)** Write the integral with the order of integration reversed:
\[
\int_{0}^{3} \int_{e^y}^{e^3} \frac{x}{\ln(x)} \, dx \, dy = \int_{A}^{B} \int_{C}^{D} \frac{x}{\ln(x)} \, dy \, dx
\]
with limits of integration
- A = [ ]
- B = [ ]
- C = [ ]
- D = [ ]
### Graphs Explanation:
- **Graph A**: A region bounded by the y-axis from 0 to 3, an arc from \((e^y, y)\) to \((e^3, 3)\), and lines along \(x = e^3\).
- **Graph B**: Similar to Graph A, but mirrored over the x-axis.
- **Graph C**: A region bounded by the x-axis from \(e^0 (=1)\) to \(e^3\), a line from \(x = e^3\) to \(3\), and an arc along \((x, \ln(x))\).
- **Graph D**: Similar to Graph C, but mirrored over the y-axis.
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