Determine $\int_{0}^{4} f(x) d x$ if \[ f(x)=\left\{\begin{array}{l} 4 x \text { for } x<2 \\ 2 x \text { for } x \geq 2 \end{array}\right. \] A. 17 B. 28 C. 20 D. 12 E. 0

Step 1 ：The function \(f(x)\) is defined differently for \(x<2\) and \(x\geq2\). Therefore, we need to calculate the integral from 0 to 2 and from 2 to 4 separately, and then add them together.

Step 2 ：For \(x<2\), \(f(x)=4x\). The area under the curve from 0 to 2 is a trapezoid with bases 0 and 8, and height 2. Using the formula for the area of a trapezoid, \((\text{base1} + \text{base2}) / 2 \times \text{height}\), we find that the area is \(8\).

Step 3 ：For \(x\geq2\), \(f(x)=2x\). The area under the curve from 2 to 4 is a trapezoid with bases 4 and 8, and height 2. Using the formula for the area of a trapezoid, we find that the area is \(12\).

Step 4 ：Adding the two areas together, we find that the integral of \(f(x)\) from 0 to 4 is \(8 + 12 = 20\).

Step 5 ：Final Answer: \(\boxed{20}\)