The image shows the expression:

$\int_{0}^{\frac{1}{2}} (8x - 2x^2) \, dx - \int_{4}^{\frac{1}{2}} (8x - 2x^2) \, dx$

We will evaluate both integrals step by step.

Step 1: Solve the indefinite integral

The function to integrate is $8x - 2x^2$. Its integral is: $\int (8x - 2x^2) \, dx = 4x^2 - \frac{2x^3}{3} + C$

Step 2: Evaluate the definite integrals

1. First integral: [ \int_{0}^{\frac{1}{2}} (8x - 2x^2) , dx = \left[ 4x^2 - \frac{2x^3}{3} \right]_0^{\frac{1}{