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Question
Math
Posted 6 months ago

Let RR be the region shown below.

Let f(x,y)=1x+2yf(x, y)=\frac{1}{x}+\frac{2}{y}.
What is Rf(x,y)dA\iint_{R} f(x, y) d A after a change of variables into polar coordinates?

Choose 1 answer:
(A)
π35π6131cos(θ)+2sin(θ)drdθ
\int_{-\frac{\pi}{3}}^{\frac{5 \pi}{6}} \int_{1}^{3} \frac{1}{\cos (\theta)}+\frac{2}{\sin (\theta)} d r d \theta

(B) π32π3131cos(θ)+2sin(θ)drdθ\int_{-\frac{\pi}{3}}^{\frac{2 \pi}{3}} \int_{1}^{3} \frac{1}{\cos (\theta)}+\frac{2}{\sin (\theta)} d r d \theta
(C) π35π6131sin(θ)+2cos(θ)drdθ\int_{-\frac{\pi}{3}}^{\frac{5 \pi}{6}} \int_{1}^{3} \frac{1}{\sin (\theta)}+\frac{2}{\cos (\theta)} d r d \theta
(D) π32π3131sin(θ)+2cos(θ)drdθ\int_{-\frac{\pi}{3}}^{\frac{2 \pi}{3}} \int_{1}^{3} \frac{1}{\sin (\theta)}+\frac{2}{\cos (\theta)} d r d \theta
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Answer from Sia
Posted 6 months ago
Solution by Steps
step 2
The function in polar coordinates becomes f(r,θ)=1rcos(θ)+2rsin(θ)f(r, \theta) = \frac{1}{r\cos(\theta)} + \frac{2}{r\sin(\theta)}
step 3
The Jacobian of the transformation from Cartesian to polar coordinates is rr, so the area element dAdA becomes rdrdθr dr d\theta
step 4
The limits for θ\theta are from 00 to π2\frac{\pi}{2} since the region R is a quarter-circle in the first quadrant
step 5
The limits for rr are from 00 to 33 because the radius of the quarter-circle is 33 units
step 6
The double integral in polar coordinates is 0π203(1rcos(θ)+2rsin(θ))rdrdθ\int_{0}^{\frac{\pi}{2}} \int_{0}^{3} \left(\frac{1}{r\cos(\theta)} + \frac{2}{r\sin(\theta)}\right) r dr d\theta
step 7
Simplifying the integrand, we get 0π203(1cos(θ)+2sin(θ))drdθ\int_{0}^{\frac{\pi}{2}} \int_{0}^{3} \left(\frac{1}{\cos(\theta)} + \frac{2}{\sin(\theta)}\right) dr d\theta
step 8
Comparing the simplified integrand with the given options, we find that none of the options match exactly because they all have incorrect limits for θ\theta and/or the integrand is not correctly expressed
None of the options A, B, C, or D are correct.
Key Concept
Change of variables in double integrals
Explanation
When changing from Cartesian to polar coordinates, the function and the limits of integration must be correctly transformed. The function f(x,y)f(x, y) becomes f(r,θ)f(r, \theta), and the area element dAdA becomes rdrdθr dr d\theta. The limits of integration for rr and θ\theta must correspond to the region of integration in polar coordinates.

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