Let R be the region shown below.
Let f(x,y)=x1+y2.
What is ∬Rf(x,y)dA after a change of variables into polar coordinates?
Choose 1 answer:
(A)
∫−3π65π∫13cos(θ)1+sin(θ)2drdθ
(B) ∫−3π32π∫13cos(θ)1+sin(θ)2drdθ
(C) ∫−3π65π∫13sin(θ)1+cos(θ)2drdθ
(D) ∫−3π32π∫13sin(θ)1+cos(θ)2drdθ
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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)=rcos(θ)1+rsin(θ)2
step 3
The Jacobian of the transformation from Cartesian to polar coordinates is r, so the area element dA becomes rdrdθ
step 4
The limits for θ are from 0 to 2π since the region R is a quarter-circle in the first quadrant
step 5
The limits for r are from 0 to 3 because the radius of the quarter-circle is 3 units
step 6
The double integral in polar coordinates is ∫02π∫03(rcos(θ)1+rsin(θ)2)rdrdθ
step 7
Simplifying the integrand, we get ∫02π∫03(cos(θ)1+sin(θ)2)drdθ
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 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) becomes f(r,θ), and the area element dA becomes rdrdθ. The limits of integration for r and θ must correspond to the region of integration in polar coordinates.
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