Evaluate ∫∫C xy dx + (x − y) dy; C consists of the line segments from (0, 0) to (2, 0) and from (2, 0) to (3, 2).

Question

Evaluate ∫∫_C xy dx + (x − y) dy; C consists of the line segments from (0, 0) to (2, 0) and from (2, 0) to (3, 2).

Answer 1

It looks like this is a line integral along the two curves that you gave:

For the first curve int from 0 to 2 of x(0)dx = 0 and the second term is 0 because y is constant (=0, but that part of the integral is constant for any horizontal line)

For the second curve we write the equation of the curve (two- point form) y =2(x-2) or x = (1/2)y+2

integral from 2 to 3 of (x(2(x-2))dx + integral from 0 to 2 of ((1/2)y+2-y)dy

(2/3)x³ - 2x² from 2 to 3 + -(1/4)y² + 2y from 0 to 2

(18 - 18) - (16/3 - 8) + (-1 + 4) - (-0 + 0) = 8/3 + 3 = 17/3

Please check the math - it's fiddly.

Please consider a tutor. Take care.