Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Instruction:**
Choose the correct region of integration \(\int_0^4 \int_0^{\sqrt{16-y^2}} \sqrt{x^2 + y^2} \, dx \, dy\).
Assume that in each figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.
**Diagrams:**
There are four diagrams with regions shaded in light blue. Each diagram represents a different quarter of a circle, centered at the origin and located in one of the quadrants.
1. **Top-left diagram:** Shaded region is the first quadrant.
2. **Top-right diagram:** Shaded region is the second quadrant.
3. **Bottom-left diagram:** Shaded region is the third quadrant.
4. **Bottom-right diagram:** Shaded region is the fourth quadrant.
**Task:**
Evaluate \(\int_0^4 \int_0^{\sqrt{16-y^2}} \sqrt{x^2 + y^2} \, dx \, dy\) by changing to polar coordinates.
(Use symbolic notation and fractions where needed.)
\[
\iint_D f(r, \theta) \, dr \, d\theta = \underline{\phantom{_________________________________________________________}}
\]](https://content.bartleby.com/qna-images/question/5c1f44d6-2912-49cf-894d-a588f3241dc5/4fb38899-4371-4298-9d82-f05de4249919/7n90n36_processed.png)
Transcribed Image Text:**Instruction:**
Choose the correct region of integration \(\int_0^4 \int_0^{\sqrt{16-y^2}} \sqrt{x^2 + y^2} \, dx \, dy\).
Assume that in each figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.
**Diagrams:**
There are four diagrams with regions shaded in light blue. Each diagram represents a different quarter of a circle, centered at the origin and located in one of the quadrants.
1. **Top-left diagram:** Shaded region is the first quadrant.
2. **Top-right diagram:** Shaded region is the second quadrant.
3. **Bottom-left diagram:** Shaded region is the third quadrant.
4. **Bottom-right diagram:** Shaded region is the fourth quadrant.
**Task:**
Evaluate \(\int_0^4 \int_0^{\sqrt{16-y^2}} \sqrt{x^2 + y^2} \, dx \, dy\) by changing to polar coordinates.
(Use symbolic notation and fractions where needed.)
\[
\iint_D f(r, \theta) \, dr \, d\theta = \underline{\phantom{_________________________________________________________}}
\]
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