in today's lesson I'm going to go over applications of integration it's basically going to be a review of the formulas associated with this topic so first let's talk about anti-derivatives the derivative of capital F is the same as lowercase f the anti-derivative of lowercase f is going to be capital f so this symbol it represents the integral sign so this is the indefinite integral of F ofx and that's going to give us the anti-derivative of lowercase f which is capital F and of course we need to add the constant of integration C now let's talk about rect to linear motion we know that the velocity function is the derivative of the position function the instantaneous acceleration function is the derivative of the Velocity function likewise the integral or the anti-derivative of the Velocity function is going to give you the position function and the anti-derivative or the indefinite integral of the acceleration function will give you the velocity function by the way you could find all of these formulas in the formula sheet that I'm going to post in the links below so now that we talked about indefinite integrals let's talk about definite integrals so this right here is an indefinite integral because you don't have the limits of integration but once you include the limits of integration the lower limit and the upper limit you now have a definite integral the result of a definite integral is a number a value a constant whereas the result of an indefinite integral is a function whether in terms of X or in terms of t or something so this might give you a value like 26 or -34 but this will give you a function so this is a definite integral and this is an indefinite integral but how can we evaluate indefinite integrals I mean I take that back how can we evaluate a definite integral we know that the indefinite integral of f ofx will give us capital F but to evaluate it once you get capital f F you need to plug in the upper limit and the lower limit of integration so this is going to be F of B minus F of a so that's how you can evaluate definite integrals first you need to find the anti-derivative and then you need to plug in the upper limit of integration and then the lower limit of integration top minus Bottom now let's talk about properties of definite integral the definite integral of a to B of f ofx DX this is equal to negative B to a f ofx DX so by switching A and B the sign will change the definite integral from a to a this is going to equal Z because F of a minus F of a they will cancel the definite integral of a constant where you don't have a variable like X or t just a number it's going to be that constant times the difference of the limits of integration so for example let's say we have the definite integral from 2 to 5 of8 DX this is going to be8 * 5 - 2 so8 * 3 that's -4 now definite integrals are additive so if we have the definite integral from A to B of f ofx DX and then plus the definite integral from B to C of f ofx DX this is going to equal the definite integral from a to c f ofx DX and if you want to understand why that's the case the first one this is going to be F of B minus F of a the second one this is going to be plus F of C minus F of B notice that F of B they cancel that one is positive this one is negative and we get the remaining result F of C minus F of a which is what we have on the right side now let's talk about how we can calculate the area under a curve the most simplest way to do this is to evaluate the definite integral of the function now sometimes you may not have a function or other times we have a function that's difficult to integrate in either case you could use a limit process to get the area under the curve so the value of the definite integral is also equal to the Limit as n approaches Infinity of the sum from I = 1 all the way to n of the function f of x sub I * Delta X so using a limit process here's what we're doing imagine we have some function F ofx and we want to determine the area on the Clos interval A to B so a is where we start B is where we end now one way in which we can approximate the area is by breaking it up into rectangles and we can calculate the area of each rectangle using something called REM and sums it could be the left end points using the right end points or even the midpoint rule Delta X is the width of each rectangle and F ofx sub I is the height of the rectangle so we have height times width that will give us the area of each rectangle and once we take the sum of all of them that will help us to calculate the area under the curve now n is the number of rectangles so for the example that I have we have a total of five rectangles so n would be five as n becomes larger and larger the approximation gets closer and closer to the exact answer so when n becomes Infinity we're going to get an exact answer of the area but when N is a small number like four or five it's an approximation but if you were to use 10 rectangles your approximation will be better if you use 20 rectangles even better so that's why we have a limit process here now Delta X is the width of each rectangle and to calculate the width it's the difference between a and b divided by the number of rectangles now in order to use this limit process by the way for those of you who want to see examples on it I'm going to post some links in the description section below where you can see how to use the limit process to calculate the area now in order to use the limit process you need to be familiar with certain summation formulas that'll make the process a lot easier to calculate here's the first one so the sum from I = 1 to n of the constant C is just C * n now if you just have I it's going to be n * n + 1 / 2 for I 2 it's going to be n * n + 1 * 2 n + 1/ 6 for I to the 3 power it's n n + 1 / 2^ 2 and you could find all of these formulas in the formula sheet that I'm going to post in the description section now for I to 4th it's going to be 6 n to the 5th + 15 n 4 + 10 n Cub - n all divided by 30 and there's another one I to 4 and I to 5th not very common but it could happen where you may need to use it for I to 5th it's 2 N 6+ 6 n 5th power + 5 n 4th power - n^ 2 over 12 so for those of you who want an example of how to use this let's say we want to find the sum of i^2 from I = 1 to 5 so this is going to be 1 2 + 2 2 + 3 2 + 4 2 + 5 2 now if this were to go up to 100 it will take a long time to calculate it but going up to five we can do this mentally so this is 1 + 4 + 9 + 16 + 25 so 1 + 9 that's going to be 10 and then 4 + 16 that is 20 and 10 + 20 is 30 30 + 25 will give us 55 now we could use the formula for i^2 it's n * n + 1 * 2 n + 1 / 6 so n is 5 so that's 5 5 + 1 is 6 2 n + 1 2 * 5 is 10 + 1 that's 11 over 6 6 will cancel and we get 5 * 11 which is 55 so as you can see these formulas are really use for for taking a sum of squared or Cub terms now let's talk about some other ways in which we can approximate the area under the curve so dealing with the topic of REM and sums if you want to calculate the area using the left end points so first let me draw a number line so let's say we want to find the area from A to B and we want to break it up into let's say four sub intervals so n is four this will be X Sub 0 x sub 1 x sub 2 x sub3 xub 4 now sometimes n could be larger so let's also call this X subn and let's call this x sub n minus1 the second before the last one so the area used in the left end points so this Five Points here but we're only going to use four because n is four so with the left end points it will be 0 1 2 and 3 so a generic equation for the area using the left end points will be Delta X that's the width of each rectangle and then we're going to add the heights of each rectangle so the first one starting from the left F of x0 plus F of X1 plus F of X2 and we're going to stop at in this case it will be F of x3 but in the general case it will be f ofx n minus one so using the left end points we start from the left we use all the end points except the last one Now using the right end points you could look at it the other way as we're starting from the right we're going to use everything except the one on the left or in other words you start from the second one you go all the way to the end so starting from the second point it's F of X1 plus F of X2 plus F of X3 F of x3 is the same as f of x n minus one well first I need to put dot dot dot to write a general equation actually it's better to write it this way F of x3 and then the last one will be F of xn so we're starting with the second point and we're going to stop at the last point now using the midpoint rule we want to get the points in the middle so5 1.5 2.5 3.5 so this is going to be F ofx .5 and then plus F ofx of 1.5 and then F ofx 2.5 and we're going to continue the pattern all the way to the last point minus5 so here's the last point x of n if you subtract that by 0.5 you'll get this point which for this example will be 3.5 so that's how you can approximate the area under the curve or even the value of a definite integral using reman sums with the left end point the right end point and the midpoint and keep in mind Delta X just like before it's B minus a/ n now there are some other forms of approximate integration that you may want to be familiar with this includes the trapeo rule and Simpson's rule so let's start with the trapeo rule T subn by the way the area which is the definite integral can be approximated using T subn just as it can be approximated with a sub L A subr and A subn but for the trapez zorder rule it's going to be Delta x / 2 * FX 0 + 2 * FX1 plus 2 * f ofx 2 and then this will continue and then you have the second to last one f ofx - 1 it's 2 * the value of that plus F ofx subn so notice that the first and the last one has a coefficient of one every other term in the middle has a coefficient of two very similar to the shape of a trapezoid the first and the last point they're at the bottom let's assign that a value of one every other point in the middle is has a value of two for the coefficient so that's the formula for the trapeo rule to estimate the area under a curve now let's talk about The Simpsons rule s subn is equal to Delta X over 3 * F ofx 0 + 4 * F of X1 + 2 * f ofx 2 and then 4 * F of x3 and the pattern is going to alternate between 2 and four for the middle terms now the third to last term is going to be 2 f ofx n minus 2 and the second to last term is going to be 4 F ofx -1 and the last term will have no coefficient in front of it so just like the trapez order rule the first and last term has a coefficient of one and then all the terms in the middle will alternate between four and two so the second and the second to last term will have a value of four but the ones in the middle will alternate between two and four with the trapeo rule all the middle ones have a coefficient of two and just like before Delta X is B minus a n so that's another way in which you can approximate the area under a curve and also the definite integral is by using The Simpsons Rule now let's move on to the fundamental theorem of calculus in short FTC part one now let's say we have some function G ofx and let's define it to be the integral from a tox of the function f of T DT in that case G Prime is going to be equal to the value of F ofx and here's why once we integrate lowercase f we'll get capital F evaluated from the top integran minus the bottom one now if we differentiate G of X so we can get G Prime of X we need to take the derivative of the results on the right the derivative of capital F is lowercase f and the derivative of a constant is zero so we can see that g Prime of X will equal F ofx so we could say that the derivative of this expression of the integral of a tox F of T DT that's going to simplify to F ofx which we shown here and that's the basic idea behind the fundamental theorem of calculus part one part two you're already familiar with it it's basically evaluating definite integrals so the integral of a to B of f ofx DX is equal to F of B capital F of B minus F of a that's part two of the fundamental theorem of calculus now the next topic is something called the net change theorem it looks very similar to part two of FTC the fundamental term of calculus the integral of the derivative of capital F is going to be F of B minus F of a that's the net change theem the difference between FTC part two is this would be just lowercase f as opposed to derivative of capital F so the basic idea is if you have some derivative function or some function that describes the rate of something and you want to find the accumulation you could use the net change strum to determine you know how much isum accumulated so let's say if you have a function that describes the rate of water flow into a tank and you want to calculate the accumulation of water um from time a to time B you could use the net change term if you have a function that describes the rate of water flow going into the tank same thing with velocity if you have velocity or how fast something is moving using the net change term you can determine the displacement of that moving object over time now let's move on I don't want to make this video too long let's talk about how we can calculate the area between two curves so we talked about how to calculate the area underneath one curve but between two curves we're going to have two functions the top function let's call it f ofx the bottom function G of X and we want to find the area between these two curves from A to B so the area of the Shaded region the area is going to be the definite integral from A to B of the top function which is f ofx minus the bottom function G ofx now sometimes you may need to calculate the area in terms of Y as opposed to in terms of X so let's say we have two functions the one on the right is f of Y the one on the left is G of Y and we want to integrate it from C to D along the Y AIS so we wanted to determine the area of this shaded region in this case the area is going to be the definite integral from C to D of the function on the right which is f of Y minus the function on the left in this case that's G of Y Dy so that's how you could determine the area between curves now sometimes you need to determine the volume of the shape generated when it's rotated about an axis so this is going to be the dis method there's also the washer method the shell method and volume by cross-sections but let's start with the dis method so let's say we have a function that looks like this and we want to rotate it along about the xais and we want to calculate the volume of the shape generated from A to B so using the dis method formula it's going to be Pi time the integral from A to B of the radius function squared so the radius is going to be the difference between the curve and the axis of rotation now this can be done in terms of X and in terms of of Y so let's say we have a function that looks like this but this time we want to rotate it about the Y AIS instead of the x axis and we want to find the volume from C to D using the dis method it's going to be Pi integral from C to D the radius squared but it's going to be in terms of Y instead of X so r y is going to be the distance between the axis of rotation and the curve now I don't want to make this video too long so for those of you who want the formulas for the washer method the shell method and volume by cross-sections in addition to other formulas that I didn't uh cover in this video like the work done by a force the mean value themm for integrals average value of a function Arc Length surface area you can find all those the formulas in the formula shape you know down below this video also I'm going to be posting other videos as well that will give you example problems on how to calculate the volume by the disc method how to calculate the area between curves how to use the shell method and the washer method so I'm going to put a lot of resources in the links down below for those of you who want more practice problems to prepare for your exam coming up so that's it for this video thanks again for watching and feel free to check out those other videos