Evaluating definite integrals on adjacent intervals
About this video
Sal introduces a value C with A <= C <= B and shows that the area under F of X from A to B is the sum of the areas from A to C and from C to B. So the integral from A to B equals the integral from A to C plus the integral from C to B. He explains this is useful for functions with jumps, like step functions, and for proving the fundamental theorem of calculus.
You'll be able to
- split a definite integral at a point C between the bounds.
- find an unknown piece of an integral using the adjacent intervals property.
- explain why this property helps with functions that jump.
Key words
- Adjacent intervals
- Intervals that share an endpoint, like [A, C] and [C, B].
- Definite integral
- The area under a curve and above the x-axis between two x-values.
- Discontinuity
- A place where a graph breaks or jumps.
- Step function
- A function that is constant on pieces and jumps between levels.
Transcript
- 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've denote it as the definite integral from A to B of F of X, DX. Now what I wanna do with 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, 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 wanna 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 I've 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 can 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 a 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 in this interval that'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 disc, 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 utilize that property. So if this is A, this is B, and let's say the function, I'm just gonna make it constant over an interval.
So 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 looked 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 here using this integral property.
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