Question

Express the integral ∫∫_R e^x² + y²dA in polar coordinates.

a) ∫∫_R eʳ² r dr dθ
b) ∫∫_R eʳ r dr dθ
c) ∫∫_R eʳ² dr dθ
d) ∫∫_R eʳ dr dθ

Answer 1

The correct expression for the integral ∫∫_R e^x² + y² dA in polar coordinates is ∫∫_R e¹² r dr dθ. Option B is the correct answer.

Step-by-step explanation:

To express the integral ∫∫_R e^x² + y² dA in polar coordinates, we replace the Cartesian coordinates (x, y) with polar coordinates (r, θ) where x = r · cos(θ) and y = r · sin(θ). The expression e^x² + y² transforms to e^(r^2) when written in polar coordinates, because x² + y² = r^2. Additionally, the area element dA in Cartesian coordinates is related to polar coordinates by dA = r dr dθ, as the area of a differential annulus with radius r and thickness dr is 2πr dr in polar coordinates.

Hence, the integral in polar coordinates should include the factor of r that comes from the area element. The correct option is b) ∫∫_R e¹² r dr dθ, where the expression e¹² includes the r² inside the exponent and the additional r comes from the area element in polar coordinates.