Why is the definite integral between negative limits of a function with positive values negative even though the area bound by the x-axis is positive? for example the integral of y=x^2 between x=-2 and x=-1

Referring back to the definition of an integral, it is the sum of small elements on the x-axis (dx) multiplied by the value of the function at that point (y) commonly expressed as the sum of ydx. Since x is negative in this region, so is dx, resulting in all of the elements of the sum being negative. A useful way of remembering this is to think about the problem graphically, and what quadrant our function crosses:x,y > 0 Both ydx (and in turn the integral in this quadrant) is positivex > 0 > y Both ydx (and in turn the integral in this quadrant) is negativex < 0 < y Both ydx (and in turn the integral in this quadrant) is negativex,y < 0 Both ydx (and in turn the integral in this quadrant) is positive
(which is case 3 for the function y=x² for negative limits)