Final answer:
The integral ∫ᵣ 8x² da over the region bounded by the ellipse is evaluated using a transformation to the uv-plane, with the correct Jacobian factor, resulting in the integral ∫ᵣ 40u² dv. The correct option is a) ∫ᵣ 40u² dv
Step-by-step explanation:
The student's question involves evaluating an integral over a region bounded by an ellipse using a given transformation. The given transformation is x = 5u and y = 6v. To evaluate the integral ∫ᵣ 8x² da, where ᵣ is the region bounded by the ellipse 36x² + 25y² = 900, we first rewrite the equation in terms of u and v using the transformation:
36(5u)² + 25(6v)² = 900, which simplifies to u² + v² = 1.
Now the region ᵣ is the unit circle in the uv-plane. To adjust for the change in variables, we need the Jacobian of the transformation, which for our transformation is Jacobian = 30 (calculated by taking the determinant of the matrix formed by partial derivatives of x and y with respect to u and v).
The integral becomes ∫ᵣ 8(5u)² 30 dvdu = ∫ᵣ 200u² dvdu. Splitting the integral over the region in the uv-plane, we realize option (a) ∫ᵣ 40u² dv is the correct form assuming we have a constant outside the integral from the Jacobian.
The correct option is a) ∫ᵣ 40u² dv