Pick a curve and a left/right boundary. The shaded region is the definite integral — the running total of the curve's height across that interval.
Split the interval between a and b into thin vertical strips, each with height equal to the curve at that point. Add up the areas of all those strips — as they get thinner and more numerous, that sum settles on a single number. That number is the definite integral, written:
It's the signed area between the curve and the x-axis: positive where the curve sits above the axis, negative where it dips below.