Question

Compute each of the following integrals by changing to polar coordinates

(a) ∬_D x^2y dA, where D is the top half of the disk centered at the origin with radius 3 in polar coordinates.

(b) ∬_D sin(x^2y^2) dA, where D is the region in the first quadrant between the circles centered at the origin of radii 1 and 4 in polar coordinates.

Answer 1

To compute the given integrals by changing to polar coordinates, express the integrand and differential area element in polar coordinates. Evaluate the integrals by determining the limits of integration. For part (a), the integral is ∫∫(0 to 3)(0 to π) r^3cos^2θsinθ r dr dθ.

Step-by-step explanation:

To compute each of the given integrals by changing to polar coordinates, we need to express the integrand and the differential area element in terms of the polar coordinates. We will use the expressions for converting between Cartesian and polar coordinates and then evaluate the integrals by determining the limits of integration.

(a) ∬_D x^2y dA, where D is the top half of the disk centered at the origin with radius 3 in polar coordinates:

In polar coordinates, x = rcosθ and y = rsinθ. Therefore, x^2y can be expressed as (rcosθ)^2(rsinθ) = r^3cos^2θsinθ. The differential area element dA in polar coordinates is given by dA = r dr dθ.

The limits of integration are r = 0 to r = 3 and θ = 0 to θ = π. We can now proceed to evaluate the integral:

∬_D x^2y dA = ∫0ˆ3 ∫0ˆπ r^3cos^2θsinθ r dr dθ.