Question

Evaluate the integral by making an appropriate change of variables. ∬ 2 sin(9x^2 + 16y^2) dA, where r is the region in the first quadrant bounded by the ellipse 9x^2 + 16y^2.

Answer 1

The student's question involves evaluating a double integral by making a change of variables to polar coordinates, which simplifies the expression and makes use of symmetry for easier integration.

Step-by-step explanation:

The student is asked to evaluate the double integral ∫∫ 2 sin(9x^2 + 16y^2) dA, where the region r is bounded by the ellipse 9x^2 + 16y^2. To approach this problem, one useful method is to make a change of variables that simplifies the integral. Polar coordinates are often useful for integrals involving symmetry and ellipses. The transformation would be x = r cos(θ) and y = r sin(θ), resulting in the new ellipse equation r^2(cos^2(θ) + sin^2(θ)) = 1, due to the fact that for an ellipse, a change of variables can be used to turn it into a circle in polar coordinates.

The differential area element in polar coordinates is given by dA = r dr dθ. This change of variables produces a new integral ∫ (from 0 to 2π) ∫ (from 0 to 1) 2r sin(r^2) dr dθ to be evaluated. Eventually, you will integrate over r first from 0 to 1 and then integrate over θ from 0 to 2π. The symmetry of the problem facilitates the integration process and simplifies the integral to a more manageable form.