Determine ${\int }_0^{4}f(x)dx$ if
$$f(x)=\{\begin{array}{l}4x\text{ for }x<2\\ 2x\text{ for }x\ge 2\end{array}$$
A. 17
B. 28
C. 20
D. 12
E. 0

Step 1 ：The function $f(x)$ is defined differently for $x<2$ and $x\ge 2$. 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 $x\ge 2$, $f(x)=2x$. 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: $\boxed{{\displaystyle 20}}$