Question

Evaluate the given double integral by using a stuitable change of coordinates. (a) ∬D ​(3x−2y) dA, where D is the region bounded by the lines y = −2x+1, y = −2x+3 ,y = 3x/2​−4 and y = 3x/2​+2. (b) ∫²₁∫ʸ₀ 1/​(x²+y²)²​ dx dy

Answer 1

To evaluate the given double integral, we can use a change of coordinates to transform the region into a simpler domain. By finding the intersection points of the given lines, we define a new coordinate system. Using the transform and Jacobian determinant, we rewrite the integral and integrate over the transformed region to obtain the final result.

Step-by-step explanation:

To evaluate the given double integral ∬D ​(3x−2y) dA, we can use a change of coordinates by transforming the region D into a simpler domain.

1. First, let's find the coordinates of the four intersection points of the given lines: (1, -1), (1, 3), (2, -1.5), and (2, 4).
2. We can use these points to define a new coordinate system in terms of u and v, where:

• u = x - y
• v = 3x/2 + y/2

The inverse transformation is given by:

• x = (2u + v) / 5
• y = (v - u) / 5

The Jacobian determinant of this transformation is 1/5.Now, we can rewrite the double integral in terms of u and v:

• ∬D ​(3x−2y) dA = ∬R ​(3((2u + v) / 5) - 2((v - u) / 5)) * (1/5) du dv

Here, R is the transformed region defined by u = 1 to u = 2 and v = -1.5 to v = 4.Simplifying the expression, we obtain:

• (3/25) ∬R (7u + 3v) du dv

Integrating with respect to u and v over the region R will give us the final result.