$\int_{0}^{2} 4 x d x=$

Step 1 ：The question is asking for the definite integral of the function 4x from 0 to 2. The integral of a function can be thought of as the area under the curve of the function. In this case, the function is a straight line, so the area under the curve is a trapezoid.

Step 2 ：The definite integral of a function can be found by finding the antiderivative of the function and then evaluating it at the upper and lower limits of integration.

Step 3 ：The antiderivative of 4x is 2x^2.

Step 4 ：Evaluate the antiderivative at the upper limit of integration (2) and subtract the value of the antiderivative at the lower limit of integration (0).

Step 5 ：The definite integral is 8.

Step 6 ：Final Answer: \(\boxed{8}\)