Final answer:
To evaluate the given integral ∬ 8 sin(25x² + 9y²) da over the region r bounded by the ellipse 25x² + 9y² = 1 in the first quadrant, we switch to the new coordinate system u = 5x and v = 3y. We then convert to polar coordinates and solve the integral step by step.
Step-by-step explanation:
To evaluate the given integral ∬ 8 sin(25x² + 9y²) da over the region r bounded by the ellipse 25x² + 9y² = 1 in the first quadrant, we need to switch to the new coordinate system u = 5x and v = 3y. This allows us to rewrite the integral in terms of u and v as ∬ 8 sin(u² + v²) (du dv/15). Next, we evaluate this integral by converting to polar coordinates. In polar coordinates, the region r is described by the inequality 25r² = 1, which simplifies to r = 1/5. Therefore, the integral becomes ∫∫ 8 sin(r²) r dr dθ over the region r = 0 to 1/5 and θ = 0 to π/2. Solving this integral step by step will give the final evaluation of the integral.