Question

Evaluate the integral by making an appropriate change of variables.

∫∫ᵣ 9 cos (5 (y-x/y+x)) dA where R is the trapezoidal region with vertices (6,0), (10,0), (0, 10), and (0,6)

Answer 1

To evaluate the given double integral, we can make a change of variables and convert it into a double integral over a new region. Then, by calculating the Jacobian and finding its determinant, we can write the new double integral and evaluate it.

Step-by-step explanation:

To evaluate the given double integral, we can make a change of variables and convert it into a double integral over a new region. Let's define a new variable u = y - x and v = y + x.

Next, we need to find the Jacobian of the transformation. The Jacobian matrix J is given by:

J = |∂u/∂x ∂u/∂y|

|∂v/∂x ∂v/∂y|

After calculating the Jacobian and finding its determinant, we can write the new double integral as:

∫∫R' 9 cos(5v) |J| dA'

where R' is the transformed region.