Question

Evaluate the double integral ∬D4xydA, where D is the triangular region with vertices (0,0),(1,2), and (0,3)

Answer 1

To evaluate the double integral ∬D4xydA, you need to break down the triangular region D into separate integrals and sum them up.

Step-by-step explanation:

The double integral ∫∫D4xydA can be evaluated by breaking down the triangular region D into two separate integrals:

1. The first integral is taken over the region where x ranges from 0 to 1 and y ranges from 0 to 2x. This integral represents the area below the line y = 2x within the triangle.
2. The second integral is taken over the region where x ranges from 1 to 0 and y ranges from 2x to 3. This integral represents the area below the line y = 3 and above the line y = 2x within the triangle.

By evaluating both integrals and summing them up, you will obtain the value of the double integral.

Answer 2

The value of the double integral ∬D 4xy dA is 81/8.

To evaluate the double integral ∬D4xydA, where D is the triangular region with vertices (0,0), (1,2), and (0,3), we can follow these steps:

Determine the limits of integration for x: x ranges from 0 to the x-coordinate of the line connecting (0,0) and (1,2), which is 3/2.

Determine the limits of integration for y: y ranges from 0 to 3.

Set up the double integral in Cartesian coordinates: ∫(0 to 3) ∫(0 to 3/2) 4xy dx dy.

Integrate with respect to x first, treating y as a constant: ∫(0 to 3/2)
`2x^(2y)`

dx =
`[(2/3)x^(3y)]` evaluated from x = 0 to x = 3/2 = (9/4)y.

Integrate the result with respect to y: ∫(0 to 3) (9/4)y

dy =
`[(9/8)y^2]` evaluated from y = 0 to y = 3 = 81/8.