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let's say that I have some function s of T which is position as a function of time position as a function function of time and let me graph a potential s of T right over here we have a horizontal axis as the time axis and let me just graph something I'll drop kind of parabola looking although I could have done in general but just to make things a little bit simpler for me so I'll draw it kind of parabola looking so that is if we call this a y-axis we could even call this Y y equals s of T as a reasonable as a reasonable way to graph our position as a function of time function and now let's think about what happens if we were to if we want to think about the change in position between two x let's say between time a let's say that's time a right over there and then this right over here is time B so time B is right over here so what would be the change in position between time a and between time B well at time B at time B we are at s of B we are at s of B position and at time a we are at s of we were at s of a position s of a position so the change in position between time a and time B let me write this down the change change in position position between and this might be obvious to you but I'll write it down between x times a and B is going to be equal to s of B this position this position s of B minus this position minus s of a minus s of a so nothing earth-shattering so far but now let's think about what happens if we take the derivative of this function right over here so what happens when we take the derivative of a position as a function of time so remember the derivative gives us the slope of the tangent line at any point so let's say we're looking at a point right over there the slope of the tangent line it tells us for a very small Ginty I'm exaggerating it visually for a very very small change in T how much are we changing in position how much are we changing in position so we write that as D s DT is the derivative of our position function at any given time so when we're talking about how the rate at which position changes with respect to time what is that well that is equal to velocity so this is equal to velocity well let me write this in different notations so this itself is going to be a function of time so we could write this this is equal to s prime of T these are just two different ways of writing the derivative of s with respect to T this makes it a little bit clearer that this itself is a function of time and we know that this is the exact same thing as velocity velocity as function of time as function of time which we will write which will real write as V of T so let's graph what V of T might look like down here let's graph it so let me let me put another axis another axis down here that looks pretty close to the original give myself some real estate so that looks pretty good and then let me try to graph a V of T so once again if this is my y-axis this is my t axis and I'm going to graph y is equal to V of T and if this really is a parabola then the slope over here is zero the slope the rate of change is zero and then it keeps increasing the slope gets steeper and steeper and steeper and so V of T might look something like this V of T might look something like this so this is the graph of y is equal to V of T now using this graph let's think if we can think of if we can conceptualize the distance the or the change in position between time a and between time B and between time II well let's go back to our Riemann sums let's think about what an area of a very small rectangle would represent so let's divide this into a bunch of rectangles so I'll try to I'll do it fairly large rectangle so we have some space to work with but you can imagine much smaller ones and I'm going to do a left Riemann sum here just because we've done those a bunch but we could do a right Riemann sum we could do a trapezoidal sum we could do anything we want so and then we could keep going all the way actually let me just do let me just do three right now let me just do three right over here and so this is actually a very rough approximation but you can imagine it might get closer but what is what is each the area of each of these rectangles trying what is it an approximation for well this one right over here you have F of a or I should say V of a so your velocity at time a is the height right over here and then this distance right over here is a change in time times delta T so the area the area for that rectangle is your velocity at that moment times your change in time what is the velocity at moment times your change in time well that's going to be your change in position so this will tell you this is an approximation of your change in position over this time then this rectangle the area of this rectangle is another approximation for your change in position over the next over the next Delta T and then you can imagine this is right over here is an approximation for your change in position for the next delta T so if you really wanted to figure out your change in position between a and B you might want to just do a Riemann sum if you want it approximate it you would want to take the sum from I equals 1 to I equals n of V of and I'll do a left Riemann sum but once again we could use the midpoint we can do trapezoids we could do the right Riemann sum but I'll just do a left one because that's what I depicted right here V of T V of T of I minus one so if this is this would be T naught would be a so this is the first this is the first rectangle so the first rectangle you use the function evaluated at T naught for the second rectangle use the function evaluated at T 1 we've done this in mole couple videos already and then we multiply it times each of the changes in time this will be an approximation this will be an approximation for our total for our total and let me make it clear where where delta T is equal to B minus a over the number of intervals we have we already know from many many videos when we looked at Riemann sums that this will be an approximation well will be an approximation for two things we just talked about it'll be an approximation for our change in position but it's also an approximation for our area so this right over here so we're trying to approximate approximate change in position change in position and this is also approximate of the area under the curve area under under the curve so hopefully this satisfies you that if you are able to calculate the area under the curve and actually this one's pretty easy because it's a trapezoid but even if this was a function if it was kind of a wacky function it would still apply that when you're calculating the area under the curve of the velocity function you are actually figuring out the change in position these are the two things well we already know what could we do to get a very to get the exact to get the exact area under the curve or to get the exact change in position well we just have a ton of rectangles we take the limit as the number of rectangles we have approaches infinity we take the limit as n approaches infinity and as n approaches infinity because it's because delta T is B minus a divided by and delta T is going to become infinitely small it's going to turn into DTS as one way to think about it and we already have notation for this this is the riemann this is it one way to think about a Riemann integral we just use the left Riemann sum once again we could use the right Riemann sum etc etc we could have used a more general Riemann sum but this one will work so this will be equal to the definite integral from A to B of V T V of T DT so this right over here is one way of saying look if we want the exact area under the curve of the velocity curve which is going to be the exact change in position between a and B we can denote it this it's the limit of this Riemann sum as n approaches infinity or the definite integral from A to B of V of T DT but what did we just figure out so remember this is another this is the we could call this the exact the exact change in position change in position between between times a and B but we already figured out what the exact change in positions between times a and B are it's this thing it's this thing right over here and so this gets interesting we now have a way of evaluating this definite integral conceptually we knew that this is the exact change in position between a and B but we already figured out a way to figure out the exact change of position between a and B so let me write all this down we have that the definite integral between a and B V of T DT is equal to is equal to s of B s of B minus s of a minus s of a where where let me write this in a new color where where s of T is the we know V of T is the derivative of s of T so we can say where s of T is the anti anti derivative derivative of V of T and this notion although I've written in a very non-traditional used I've used position velocity this is the second fundamental theorem of calculus fundamental theorem and you're probably wondering about the first we'll talk about that in another video but this is a super useful way of evaluating definite integrals and finding the area of look under a curve second fundamental theorem of calculus very closely tied to the first fundamental theorem which we won't talk about now so why is this such a big deal well let me write it in the more general notation the way that you might be used to seeing it in your calculus book it's telling us that if we want the area under the curve between two points a and B between two X points a and B of f of X and so this is how we would denote the area under the curve between those two intervals so let me draw just to make it clear what I'm talking about in general terms so this right over here could be f of X and we care about the area under the curve between a and B if we want to find this exact area under the curve we could we can figure it out by taking the antiderivative of F and let's just say that capital f of X is the antiderivative anti derivative or is an antiderivative because you can have multiple that are shifted by constant is an antiderivative is antiderivative of F then you just have to take evaluate the antiderivative at the endpoints and take the difference so you take the the endpoint first I guess the you subtract the antiderivative evaluated at the starting point from the antiderivative evaluated at the endpoint so you get capital F of B minus capital F of a so if you want to figure out the exact area of the curve you you you take the antiderivative of it and evaluate that at the endpoint at the endpoint and from that you subtract the starting point so hopefully that makes sense in the next few videos we'll actually apply it

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