Let's analyze and solve the given problem step by step.

The expression is: $\int_0^{\frac{1}{2}} (8x - 2x^2) \, dx - \int_4^{\frac{1}{2}} (8x - 2x^2) \, dx$

Step 1: Solve the first integral

For $\int_0^{\frac{1}{2}} (8x - 2x^2) \, dx$:

1. Split the integral: $\int_0^{\frac{1}{2}} (8x - 2x^2) \, dx = \int_0^{\frac{1}{2}} 8x \, dx - \int_0^{\frac{1}{2}} 2x^2 \, dx$

2. Solve each term:

   • For $\int_0^{\frac{1}{2}} 8x \, dx$: $\int 8x \, dx = 4x^2 + C$ Evaluate from 0 to $\frac{1}{2}$: $4\left(\frac{1}{2}\right)^2 - 4(0)^2 = 4\left(\frac{1}{4}\right) = 1$

   • For $\int_0^{\frac{1}{2}} 2x^2 \, dx$: $\int 2x^2 \, dx = \frac{2}{3}x^3 + C$ Evaluate from 0 to $\frac{1}{2}$: \