2. Calculate the following double integral: I ² = √√√√₁₂ (²^² + where the region R is defined as follows: 2xy : (x² + y²) (1 + x² + y²) drdy R = {(x, y) = R² : x ≥ 0, y ≥ 0, x ≤ x² + y² ≤ 2x} (a) Obtain the equations of the curves involved and sketch the region R. (b) Perform the integration in polar coordinates. (c) State whether, in your opinion, it is better to proceed with an integration by ver- tical or horizontal lines, should one consider integrating in Cartesian coordinates. Both methods could also be equally applicable and there could be no "better" choice. Elaborate on your reasoning in any case. (d) Write down the integral with its limits in the case you consider "better" (integra- tion by vertical or horizontal lines), without performing the integration.
2. Calculate the following double integral: I ² = √√√√₁₂ (²^² + where the region R is defined as follows: 2xy : (x² + y²) (1 + x² + y²) drdy R = {(x, y) = R² : x ≥ 0, y ≥ 0, x ≤ x² + y² ≤ 2x} (a) Obtain the equations of the curves involved and sketch the region R. (b) Perform the integration in polar coordinates. (c) State whether, in your opinion, it is better to proceed with an integration by ver- tical or horizontal lines, should one consider integrating in Cartesian coordinates. Both methods could also be equally applicable and there could be no "better" choice. Elaborate on your reasoning in any case. (d) Write down the integral with its limits in the case you consider "better" (integra- tion by vertical or horizontal lines), without performing the integration.
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:2. Calculate the following double integral:
2xy
I
= | | )
(x² + y²) (1 + x² + y²)
where the region R is defined as follows:
drdy
R = {(x, y) = R² : x ≥ 0, y ≥ 0, x ≤ x² + y² ≤ 2x}
(a) Obtain the equations of the curves involved and sketch the region R.
(b) Perform the integration in polar coordinates.
(c) State whether, in your opinion, it is better to proceed with an integration by ver-
tical or horizontal lines, should one consider integrating in Cartesian coordinates.
Both methods could also be equally applicable and there could be no "better"
choice. Elaborate on your reasoning in any case.
(d) Write down the integral with its limits in the case you consider "better" (integra-
tion by vertical or horizontal lines), without performing the integration.
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