Question

Evaluate the given integral by changing to polar coordinates. e⁽⁻ˣ² ⁻ ʸ²⁾ da, where d is the region bounded by the semicircle x = 25 - y² and the y-axis. Use the given transformation to evaluate the integral. (20x + 15y) da, where r is the parallelogram with vertices (-3, 12), (3, -12), (5, -10), and (-1, 14); x = (1/5)(u + v), y = (1/5)(v - 4u)

Answer 1

To evaluate integrals in polar coordinates, substitute the Cartesian coordinates with their polar equivalents and adjust the da element to r dr dθ. Set the limits of integration according to the given region and perform the integration. For integrals with transformations, apply the corresponding changes and use the Jacobian for the variable change.

Step-by-step explanation:

To evaluate the given integral in polar coordinates, the integral expression e⁽⁻² ⁻²⁽ da can be rewritten using the polar coordinate substitution; where x=r cos(θ) and y=r sin(θ), and da becomes r dr dθ. The bounds for r and θ need to be determined based on the given region. The semicircle x = 25 - y² can be translated into polar as r = 25 - (r sin(θ))² since y = r sin(θ), and the y-axis corresponds to the line θ = π/2 and θ = 3π/2.

As for the second part that involves the transformation (x = (1/5)(u + v), y = (1/5)(v - 4u)) and the integral of (20x + 15y) da over a parallelogram region, this requires setting up the integral bounds in the u-v plane according to the vertices given. Once the bounds for u and v are established, the integral is expressed in terms of u and v, using the Jacobian determinant for the change of variables from (x, y) to (u, v) coordinates.