Definite integrals on adjacent intervals

so we've depicted here the area under the curve f of X above the x-axis between the points x equals a and x equals B and we denote it as the definite integral from A to B of f of X DX now what I want to do in this video is introduce a third value C that is in between a and B and it could be equal to a or it could be equal to B so let me just introduce it right like just like that and I could write that a is less than or equal to C which is less than or equal to B and what I want to think about is how does this definite integral relate to the definite integral from A to C and the definite integral from C to B so let's think through that so we have the definite integral from A to C of f of X actually we have already used that purple color for the function itself so we use green so we have the integral from A to C of f of X DX and that of course is going to that's going to represent this area right over here from A to C under the curve f of X above the x-axis so that's that and then we could have the integral from C to B of f of X DX and that of course is going to represent this area right over here well the one thing that probably jumps out at you is that the entire area from A to B this entire area is just the sum of these two smaller areas so this is just equal to that Plus that over there and once again you might say why is this integration property useful that if I found a C that is that is in this interval that it's greater than or equal to a and it's less than or equal to B why is it useful to be able to break up the integral this way well as you'll see this is really useful it can be very useful when you're doing discs when you're looking at functions that have discontinuities if they have step functions you can break up the larger integral into smaller integrals you'll also see that this is useful when we prove the fundamental theorem of calculus so in general this is actually a very very very useful technique let me actually draw an integral where it might be very useful to to utilize that property so if this is a this is B and let's say the function I'm just going to make it constant over an interval C it's constant from there to there and then it's and then it drops down from there to there let's say the function look like this well you could say that the larger integral which would be the area under the curve it would be all of this let's just say it's a gap right there or it jumps down there so this entire area you can break up into two you can break up into two smaller areas so you could break that up into that area right over there and then this area right over you here using this integral property