Given:
The integral I = f0i 2 (Ž)2 dz is evaluated along the real axis from 0 to 2 and vertically upward to (2 i).
We are asked to find the value of A when I = A Bi.
Assume Ž is the complex conjugate of z.

To Find:
The value of A in the expression I = A Bi.

Solution:

Step 1: Defining the Integral:
Let's start by defining the integral:
I = f0i 2 (Ž)2 dz

Step 2: Evaluate the Integral:
We are given that the integral is evaluated along the real axis from 0 to 2 and vertically upward to (2 i). Therefore, we need to split the integral into two parts and evaluate each part separately.

Step 2a: Integral along the real axis from 0 to 2:
For this part, z is a real number. Therefore, the complex conjugate of z, Ž, is equal to z itself.
So, we can rewrite the integral as:
I1 = f02 2 (z)2 dz

Now, let's evaluate this integral. Taking the antiderivative of (z)2, we get:
I1 = [1/3 * z3]0^2
I1 = (1/3 * 2^3) - (1/3 * 0^3)
I1 = 8/3

Step 2b: Integral vertically upward to (2 i):
For this part, z is a complex number. Therefore, the complex conjugate of z, Ž, is different from z.
Let's consider z = x + yi, where x and y are real numbers. The complex conjugate of z, Ž, is then given by:
Ž = x - yi

Now, let's rewrite the integral as:
I2 = f0^2i 2 (x - yi)2 dz

To evaluate this integral, we can use the fact that dz = dx + idy.
Therefore, the integral becomes:
I2 = f0^2i 2 [(x - yi)2 (dx + idy)]

Expanding the expression, we get:
I2 = f0^2i 2 [x2 - 2xyi - y2 (dx + idy)]

Now, let's evaluate this integral. We need to integrate with respect to both x and y separately.
Integrating the real part (x2 - y2) with respect to x, we get:
f(x2 - y2) dx = [1/3 * x3 - y2x]0^2
= (1/3 * 2^3 - y2 * 2) - (1/3 * 0^3 - y2 * 0)
= 8/3 - 4y2

Integrating the imaginary part (-2xyi) with respect to y, we get:
f-2xyi dy = -2xiy

Now, let's substitute the limits of the integral. Since we are integrating vertically, the limits for y are 0 to 2.
Therefore, the integral becomes:
I2 = -2xi