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Evaluate ∬ 8 sin(25x² + 9y²) da, where r is the region in the first quadrant bounded by the ellipse 25x² + 9y² = 1; u = 5x, v = 3y.

User Klodian
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1 Answer

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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.

User Piotr Dawidiuk
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7.6k points
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