The graph of is a curve in the x-y plane. How long is that curve? A definite integral needs endpoints and we specify x = 0 and x = 4. The first problem is to know what "length function" to integrate.
Here is the unofficial reasoning that gives the length of the curve. A straight piece has . Within that right triangle, the height is the slope times . This secant slope is close to the slope of the curve. Thus is approximately .
Now add these pieces and make them smaller. The infinitesimal triangle has . Think of as and integrate: