Evaluate the double integral: ∫023∫𝑦/43/21−𝑥2𝑑𝑥𝑑𝑦using change of coordinates. Group of answer choices1/67/62/31/3 PreviousNext

To evaluate the double integral ∫ from 0 to 2/3 ∫ from y/4 to 3/2 1−𝑥^2 dx dy using change of coordinates, follow these steps:

Step 1: Identify the limits of integration The limits of integration are 0 to 2/3 for the outer integral and y/4 to 3/2 for the inner integral.

Step 2: Change of coordinates We can use polar coordinates to simplify the integral. Let x = rcos(θ) and y = rsin(θ). The Jacobian determinant for this transformation is r.

Step 3: Change the limits of integration The limits of integration in polar coordinates are 0 to 2π for θ and 0 to √(1 - r^2) for r.

Step 4: Substitute the new variables into the integral The integral becomes ∫ from 0 to 2π ∫ from 0 to √(1 - r^2) r dr dθ.

Step 5: Evaluate the inner integral The inner integral evaluates to [1/2 * r^2] from 0 to √(1 - r^2), which simplifies to 1/2 * (1 - r^2).

Step 6: Substitute this result into the outer integral The outer integral becomes ∫ from 0 to 2π 1/2 * (1 - r^2) dθ.

Step 7: Evaluate the outer integral The outer integral evaluates to [1/2 * θ - 1/6 * θ^3] from 0 to 2π, which simplifies to π - 8π^3/3.

Please note that this is a general approach and the actual computation might differ based on the specific function and limits of integration.