Another complex line integral question

[randybryan]

I have to integrate |z|²dz from 0 to 1 + 2i using the indicated paths. The first path is a straight line from the origin to 1 + 2i and the second has two lines, the first going from 0 to 2i along the y-axis and then from 2i to 1 + 2i, a line parallel to the x axis.

For the first path, the straight line, I used the parameters z = t +2ti and dz = (1 +2i)dt

|z|² = 5t² so the integral became \int 5t^2 (1+ 2i) dt between t=0 and t=1

The answer to this integral is 5/3 (1 + 2i), which is correct according to the back of the book.

However I can't get the line integral along the broken paths to generate the same answer. Can anyone spot an obvious mistake?

For the first path, x= 0 so z = 2it and dz = 2idt

|z|² = 4t² so the integral becomes \int 8it^2dt between t = 1 and t=0, giving 8/3 i

For path 2, y = 2i, so dy =0 and it just varies along x.

let z = t + 2i dz= 1 and vary from t= 0 to t= 1

|z|² = t² + 4

so the integral becomes \int (t^2 + 4)dt between 0 and 1 and the answer is t²/3 + 4t, between t=0 and t=1 which gives 13/3

now 13/3 and 8/3i does not equal 5/3 (1 +2i)

Where have I gone wrong? And I'm sure I've done something embarrassingly stupid

 

[fresh_42]

The contour integrals of $\int_C |z|^2\,dz$ depend on the paths!