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Current time:0:00Total duration:2:31

Definite integral over a single point

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.A (LO)
,
FUN‑6.A.1 (EK)
,
FUN‑6.A.2 (EK)

Video transcript

we've already taken definite integrals and we've seen how they represent or they're they denote the area under a function between two points and above the x-axis let's do something interesting let's think about a definite integral of f of X DX so it's the area under the curve f of X but instead of it being between two different X values say a and B like we've seen multiple times let's say it's between the same one let's say it's between C and C where let's say C is right over here what do you think this thing right over here is going to be equal to what does this represent what does this equal to it I encourage you to pause the video and try to think about it well if you try to visualize it you're thinking about okay a the area under the curve f of X above the x-axis from C from x equals C to x equals C so this region I guess we could call it that we think about it does have a height the height here is f of C but what's the width well there is no width it's we're just at a single point we're not going from C the C plus some Delta X or C plus some even very small change in X or some other or some C plus some other very small value we're just saying at the point C so we really were when we talking about area we're thinking about a we're thinking about things we're thinking about how much two-dimensional space you're taking up but this idea this is just a one-dimensional this is just a one-dimensional I guess you could think of it as a line segment what's the area of a line segment well a line segment has no area so this thing right over here is going to be equal to zero now you might say okay I get that I see why that could make sense what that makes intuitive sense I have I'm trying to find the area of a rectangle or I know it's height but it has its width is zero so that area is going to be zero is one way to think about it but Sal why are you even pointing this out to you to me and as we'll see especially when we can do more complex definite integration problems and solving things sometimes sometimes recognising this will help you simplify an integral wrote an integration problem dramatically or you could work to being able to get to a point like this so that you can cancel things out or you can say hey oh that thing right over there is just going to be equal to zero