MediumCalculusca-075

Change of Variables Integral

Problem

Integrate the linear function f(x, y) = a*x + b*y over the parallelogram spanned by the vectors u = (u1, u2) and v = (v1, v2) from the origin.

The substitution (x, y) = s*u + t*v maps the unit square to the parallelogram with Jacobian determinant |u1*v2 - u2*v1|. Since f is linear, the average over the unit square is (f(u) + f(v)) / 2.

Hint

Multiply the average value of f on the unit square by the absolute Jacobian determinant.

Test cases

case 1

in: [1,0,1,0,0,1]

expected: 0.5

case 2

in: [1,0,2,0,0,3]

expected: 6

case 3

in: [1,1,1,1,1,-1]

expected: 2

case 4

in: [0,1,1,0,0,2]

expected: 2

Starter codePython

def change_of_variables(a, b, u1, u2, v1, v2):
    # Your code here
    pass

4 test cases · run your code against them in the editor