Question

Evaluate the given integral by changing to polar coordinates: ∬r(cos(√(x² + y²))) dA, where r is the region in the first quadrant enclosed by the circle x² + y².

a) ∫∫r cos(θ) rdrdθ
b) ∫∫r cos(θ) dA
c) ∫∫ cos(θ) rdrdθ
d) ∫∫ cos(θ) dA

Answer 1

The correct evaluation of the integral changes the given function into polar coordinates, which results in the integral ∫∫rcos(θ) rdrdθ, encompassing the region from r = 0 to R and θ = 0 to π/2.

Step-by-step explanation:

The student has asked to evaluate the given integral by changing to polar coordinates. The integral is ∫∫r(cos(√(x² + y²))) dA, where r is the region in the first quadrant enclosed by the circle x² + y². To convert this to polar coordinates, we recognize that x² + y² = r² in polar coordinates, and dA in polar coordinates is r dr dθ. Hence, the integral becomes ∫∫(rcos(√r²) rdrdθ) which simplifies to ∫∫rcos(θ) rdrdθ, since the √r² is just r.

To find the limits of integration for r, we consider the entire region within the circle, which varies from 0 to R (the radius of the circle). For θ, being in the first quadrant, the limits are from 0 to π/2. This leads to the evaluation of the integral: ∫π/20 ∫R0 r cos(θ) dr dθ.