Area Under a Curve - Video Tutorials & Practice Problems

Area Under Linear Functions

Welcome back, everyone. So over the course of these next couple of videos, we're going to be learning an important skill to have in this course, which is being able to find the area underneath a function. Now for this video, we're specifically going to be focusing on linear functions, and if all of this sounds scary, don't sweat because it turns out finding the area under linear functions is very straightforward because we can use shapes that we're already familiar with, like the area of rectangles or triangles. So let's go ahead and just jump right into an example of this to see what these types of problems look like. So in this example, we are given the function f of x is equal to x, and we are asked to find the area under the curve from x equals 0 to x equals 5. Now I can see 0 is right about here on the x axis, and 5 is right there. Now if I want to find the area underneath this curve, all I'm really needing to do is figure out what the familiar shape is that this makes. And I can see that we clearly have a triangle which forms right here, and the area of a triangle is 1 half base times height. So the area is going to be 1 half the base of this triangle, which I can see we go from 0 to 5 times the height which goes from 0 to 5 on the y axis. So we're gonna have 1 half 5 times 5, which is 1 half of 25, and 1 half of 25 is 12.5. So this right here is going to be the area underneath this function curve, and that's all there is to it. So as you can see, whenever you're asked to find the area under a curve, what you're really being asked to do is find the area between the function's graph and the x axis, and that's exactly what we did. Now it's also possible you'll see situations where rather than just using one familiar shape, we actually have to use a combination of multiple familiar shapes. And to make sure we can understand some more of these complicated problems, well, let's just try a couple more examples. Now for each of these examples, we need to calculate the area underneath the curve for the interval that we're given. Now we'll go ahead and start with example a. Now example a asks us to find the area underneath this function from x equals negative 8 to x equals 4. I can see negative 8 is right here on the x axis, and I can see that positive 4 is up here. So this is the area that we're trying to calculate. Now to find this area, well, I could try to find a trapezoid or some kind of familiar shape here, but I think the more simple thing to do is going to be to split this up into 2 shapes that we're familiar with. Notice if I divide the shape in this fashion, we get 2 areas, area 1 and area 2. And area 1, well this is a triangle and area 2 is a rectangle. So all we need to do is add these areas together, and that's going to give us the total area underneath this curve for the interval we have. So the total area is going to be the area of a triangle, which is 1 half base times height, plus the area of a rectangle, which is just base times height. Now looking at this, I can see that the base of our triangle is going to be this distance right here. We go from negative 8 to 0, so this is a distance of 8. And the height is going to be up here on the y axis which is also 8. Now what I'm gonna do is add this to the base of this rectangle which goes from 0 to 4 times the height which goes from 0 to 8. So we have this as our math right here, and I just need to multiply and add everything together. Now one half of 8 is 4, and 4 times 8 is 32. Then we're going to add this to 4 times 8 which is 32, and 32 +32 is 64. So this right here is the area underneath this curve, and that is the solution to example a. But now let's go ahead and try example b. Example b asks us to find the area underneath the curve from x equals negative 6, which is right there, to x equals 8, which is right here. So this is the interval that we're given in this problem, but notice when we try to find this area, we're actually going to get an area above the curve if we're looking between the function and the x axis. So this is actually something that's possible to calculate? Well, it turns out that it is. In fact, the whole under the curve statement can actually be misleading, because when the function goes below the x axis, the area is going to be above the x axis, meaning the area we calculate is going to be negative. Now let's see if we can actually understand why this area becomes negative. Well, looking at this, I can see we have a rectangle that forms, so we only really need one shape here. Now what I can see is that the area of a rectangle is base times height, and what I first need to do is calculate the base of this rectangle. Well, the base is going to be this distance, which goes from negative 6 to 0, so that's a distance of 6, plus this distance, which goes from 0 to 8. 6 +8 is 14, so that is going to be our base. Now the height is going to be from here down there, but notice we're actually going down to find our height. We haven't had to do that yet. So since our height goes from 0 to negative 6, that means the height is going to be negative 6. So we have 14 times negative 6, which gives us an area of negative 84. So this is how you can calculate the area underneath this function curve, or in this case it would be the area above the function curve. So this is the strategy for finding the area between a linear function and the x axis. Hope you found this video helpful, and let's try getting some more practice.

Find the area under the curve in the function below from $x=-4$ to $x=8$.

• A. 16
• B. 32
• C. 24
• D. 40

Find the area under the curve in the function below from $x=-2$ to $x=8$.

• A. 34
• B. 38
• C. 50
• D. 42

Approximating Area Under the Curve

Hey, everyone. So in a recent video, we talked about how you could find the area under the curve of a function if you have some kind of linear function. Well, here's my question for you. How would you go about calculating the area underneath a function curve if you have a more curvy graph like this one? Well, one of the things that might be tempting is to try and draw some kind of familiar shape, but there really isn't a familiar shape that's going to work. Because if I try drawing a triangle, notice how this is not going to capture the area here, because this is not the exact shape we're dealing with. If I try drawing a rectangle, well, this shape is not even close. So how can I go about doing this? Because it's clear that the familiar shape strategy does not work for these more curvy graphs. Well, what I'm going to be discussing with you in this video is our strategy for approximating the area underneath these more curvy functions. So without further ado, let's get right into things. Now the strategy for approximating this area is by taking it and cutting it up into rectangles. Now rectangles are a familiar shape, and we know that the area of a rectangle is pretty straightforward, base times height. So if you take this shape here, this curvy graph, and you fill it with rectangles, you can find the base and the height of each rectangle, and then add everything together, and doing that will give you an approximation for the area underneath the function. So let's just go ahead and try this example here to see what this looks like. We're asked to use 5 rectangles to approximate the area underneath the function f of x is equal to x squared, and we're asked to do this on the interval from x equals 0 to x equals equals 5 with left and right endpoints. Now we're going to start with the left endpoints. And all this means when you're dealing with left end points is it means you want to start with the upper left corner, the top left corner of each rectangle and touch it to the function curve. This will make sense as we go ahead and draw these rectangles. Now my first step before actually drawing anything on our graph though, is going to be to figure out what the base is, and the base of each rectangle can be found with this equation right here. So it's going to be that our base is equal to the high point on the x axis, which we can see is b or 5, minus the lowest point on the x axis, which we can see here is 0, all divided by n, which is the number of rectangles, and we can see from our example that that's 5. So we have 5 minus 0 divided by 5, which is equal to 1. So that means that our base is 1. And this actually makes sense because if we're filling our curve here with 5 rectangles, and we want to find the width or the base of each of those rectangles, it makes sense that it would be 1, since we need to evenly split it up from 0 to 5 on the x axis. So let's go ahead and now draw in these rectangles. But remember, we're using the left endpoints or the top left corners. So my first rectangle is gonna look something like this. I'll touch it to the top left corner here, and it's going to go right there. So it's really just kind of a line touching the x axis because we're starting with left endpoints. But if I draw my second rectangle, again, these all have a base of 1, so we're going to start here on our curve and it's going to look like that. That's our second rectangle. Then our 3rd rectangle is going to touch at its top left corner right there. Our 4th rectangle is going to touch at the top left corner right about there. And then our 5th rectangle is going to touch at the top left corner here. So this is going to be what 5 rectangles underneath this curve looks like. And what I need to do is find the area for each one of these rectangles, Because what I can do here is if I'm able to find the area, which we know is just base times height, I could find the base times the height of the first rectangle plus base times the height of the second rectangle, and I could just keep adding these on until I get all these rectangles. So let's go ahead and do this. Now we can already see here that the base is 1, so we're gonna have a base of 1, but what's the height of this first rectangle? Well the height is going to be this point right about there, and we can see if we look here we're at 0. 0 squared on the y axis here would just be 0, so that means our height is 0. Now let's move on to the second rectangle. We can see that the base is 1, that's going to be the same for all these rectangles, but the height here, well the height is going to be right about at that point. And I can see it's going to be 1 squared and 1 squared is 1, so that's going to be our height. Now let's move to the 3rd rectangle. The 3rd rectangle has a base of 1 and then the height, well, we can just look right about here. For the 3rd rectangle, it's going to be 2 squared which we know is 4, so that's our height. So now we'll move to the 4th rectangle. The 4th rectangle has a base of 1 and the height we can see is going to be 3 squared, which is 9. Now let's add the 5th rectangle. The 5th rectangle is right there, it has a base of 1, and the height is going to be 4 squared, which is 16. So this right here is the approximate area underneath our function. And if you go ahead and do the math here, where you multiply all these numbers and add them together, you should get that your area is approximately equal to 30. So this is how you can approximate the area underneath a function curve using these rectangles. Now we talked about how this was a left endpoint approximation, but the question becomes, what would happen if we were dealing with right endpoints? Well, for a right endpoint approximation, it's the same exact idea, except rather than starting at the top left corner, you're now going to start at the top right corner, which we can see is already the case on this graph here. So what we need to do is find the area of each one of these rectangles to do our right endpoints. So let's go ahead and do this. Well, we know that our base is already equal to 1, and this is gonna be the same in both these cases since we're using 5 rectangles and our interval goes from 0 to 5. So our base is going to be 1, but the height is going to be a little bit different, because notice that our height is going to start on the right side this time. So height is going to be 1 squared, which is 1. Now we're going to add this to our next rectangle, the base is always 1, but the height, you can see here, is going to be 2 squared, which is 4. Now we can go ahead and add our next rectangle. The base is going to be 1, but the height of this next rectangle is going to be 3 squared, which is 9. Now we'll add our next rectangle, the base is 1, but the height is going to be 4 squared, which is 16, and then we'll go ahead and add our last rectangle. The base is 1, and the height is going to be 5 squared, which is 25. So this is what we end up getting when we use right endpoints, and if you go ahead and multiply out these numbers and add everything together, you should get that your area is approximately equal to 55. So this right here is the approximate area underneath the curve using right endpoints, and notice the two answers we got here. One of these answers gave us a lower number, but that makes sense because this is actually an under approximation. Notice there's a bunch of areas here that we actually haven't accounted for, so so that's why we got this lower number. Whereas for this case, we got a higher number, but we actually did an over approximation, because notice these rectangles have some extra area that comes over the curve that we actually don't want to account for if we're trying to get an accurate approximation. So a lot of times people will try to take these low and these high numbers and kind of average them, or figure out some number between to see how accurate they can get this area. But either way you wanna go about it, this is the strategy for approximating the area underneath a function curve when you don't have a nice linear function like a line or a constant. So hope you found this video helpful. Let me know if you have any questions, and let's move on.

Use 3 rectangles to approximate the area under $f\left(x\right)=x^2$ from $x=2$ to $x=5$ with right endpoints. 

• A. 50
• B. 40
• C. 30
• D. 35

Use 6 rectangles to approximate the area under $f\left(x\right)=x^3$ from $x=0$ to $x=3$ with left endpoints. 

• A. $\frac{225}{16}$
• B. $\frac{225}{8}$
• C. $\frac{81}{4}$
• D. 81

Approximating Area Algebraically Example 1

Let's try solving this example. So here we're asked to use 4 rectangles to approximate the area under this function, f of x is equal to 4 x squared from x equals 0 to x equals 8. And they want us to use left endpoints to solve this problem. So how can we do this? Well, whenever you are not given a graph for these situations where you need to use rectangle approximations, you're going to need to use this equation right here to solve the problem. But thankfully, it's pretty straightforward, or at least very related to what we've already done, where we just add a bunch of rectangles. So let's see what this looks like. Now what I'm first going to do is calculate the base. I think this is always a good starting point with these types of problems since the base stays constant for every rectangle that you have. Now the base is going to be this equation right here, b minus a over n. Now b, I can see is the end of the interval, which is 8. A is the start of the interval, which is 0, and n is the number of rectangles, which is 4. So we're going to have 8 minus 0 over 4, which is equal to 2. So this is going to be the base for every rectangle that we have. Now, at this point, what I'm trying to do is set up this equation. So I can do this by recognizing this is the sum of every rectangle we have. We have 4 rectangles, So our area is going to be this base of 2 times the height of each rectangle. So we'll have base times height, which is 2 times f of x 1, that's our first rectangle, plus base times height, 2 times f of x 2, our second rectangle, plus base times height of our 3rd rectangle, f of x 3, plus base times the height of our 4th rectangle, f of x 4. So these are all of our rectangles added up. Now remember, there's actually a pattern when it comes to these types of rectangles. When you're using left endpoints, x one is going to be the a value, which I can see is 0, plus 0 times delta x. Now delta x is equal to 2, but anything times 0 is just 0, so x one is just going to be 0. Then we'll have x 2, which is 0, the a value, plus 1 times delta x. Now delta x is equal to 2, so we'll have 1 times 2, which is 2. Then we'll have x 3. X 3 is going to be 0 plus 2 times delta x. Now 2 times delta x, well, that's going to be 2 times 2, which is 4. And then we'll have x 4, which is 0 plus 3 times delta x. 3 times 2 is 6, so we're going to have 0 plus 6, which is just 6. So this right here is all of the x values that we have. We have x 1, x 2, x 3, and x 4. So now that we have all of these calculated over here, well, now I can apply them to this equation. So we're going to have 2, the space stays the same, 2 times f of x 1, which is that x one was 0. So we have 2 times f of 0, plus 2 times f of 2, plus 2 times f of 4. I'm just using the x 1, x 2, x 3, x 4. We calculated it over here. Plus 2 times f of 6. Now at this point, what we need to do is take all of these inputs, plug them into our function, and figure out what the output is going to be. So we're going to have 2 times 0 plugged into our function. So it's gonna be 0 squared times 4, which is 0. Then we're going to have plus 2 times 2 plugged into our function, so that's 2 squared, which is 4. 4 times 4 is 16. Now we're going to have 4 plugged into our function. So that's gonna be 2 times 4 squared, which is which is 16 times 4, which is 64. And then we're going to have 6 plugged in to our function. So we're going to have 6 squared, which is 36 6 times 4, which is a 144, so we're going to have plus 2 times a 144. So this is what we end up getting. Now at this point, all you need to do is multiply out all the numbers and add everything together, and you should be able to do this. You could check this on a calculator or do this by hand, and you should get that your area is equal to 448. So this is the approximate area underneath the curve using left endpoints. So this is how you can solve these types of problems where you don't have a graph, but need to use the equation to approximate the area under a curve. Hope you found this video helpful, let's move on.

Using Limits to Find Area Under the Curve

Everyone, and welcome back. So up to this point, we've spent a lot of time figuring out how we can approximate the area underneath the curve of a function by filling it with rectangles and adding the area of each rectangle. Now in this video, we're going to learn how we can find the exact area underneath the curve. So we're no longer approximating. Now the results we get are going to be exact using this process of filling it with a bunch of rectangles. And this might sound like it's impossible to do, but it turns out there is a process that allows us to do this with what we've learned. So pay close attention, because I'm gonna walk you through these steps and show you how to solve these types of problems in this course. So let's get right into things. Now the main thing we need to think about is how exactly could we use rectangles to find the exact area. Well, we know that for approximating with rectangles, we're in essence just filling this up and calculating the area of each rectangle, base times height. So let's say that I were to try do doing an approximation with 2 rectangles. Well, that would look something like this, where we'd have one rectangle right here and another rectangle right there. I'm using right endpoints to do this. Notice, if I were to try and find this area, we would get a severe over approximation because we have all this extra area above the curve that's sticking out, and this would give us an overapproximation for what we have. But now let's say I wanted to fill this with more rectangles, So rather than using 2, we use 4. So 4 rectangles would look something like this. And again, I'll use these right endpoints to approximate. So we'd have 1 rectangle here, 1 rectangle there, one rectangle right about there, and one rectangle right here. Now notice when doing this, we still get an over approximation. Because if you look, we have all this extra area that's still above the curve, but this is actually a better approximation than the area we had before. So notice as we draw more rectangles as n gets bigger, the width or the base of these rectangles, I should say, gets smaller, and we get a more accurate approximation. So what exactly could we use to get the most accurate approximation possible? Well, that just means we need to increase the number of rectangles. Let's say that we had a 100 or a 1000 or even a 1000000 rectangles. This would give us a very accurate approximation. We would have very little error, very little extra area popping up above the curve. So if we could somehow find a way to get this n value, this number of rectangles to approach infinity, that would allow us to get a truly accurate approximation or the exact amount. And in this case, we actually can do that because we can't necessarily just say, oh, we have an infinite number of rectangles, but what we can do is apply a limit to this equation that we've already been using, and say, what happens if this limit approaches infinity, the number of rectangles? Well, in that case, we're going to get the most accurate approximation we can, meaning that that's going to be the exact area underneath the curve. So let's actually see how we can use this equation to solve the type of problem where we want to find the exact area. So in this example down here, it says find the exact area underneath the function f of x is equal to 1 half x from x equals 0 to x equals 6, that's our interval, using the limit definition. So this is the limit definition we have right up here, so let's see how we can piece this equation together. Now we have this argument out in front right here. We have the this limit as n approaches infinity, and then we have this summation. All this is saying is this is the summation that says we need to add a certain number of rectangles, and this time we're going to an infinite number of rectangles. So we have this argument out in front. Now what I need to find is delta x to go in here. Now we know how to calculate delta x, it's this equation that we've been using, b minus a over n. So delta x is going to be b, which is the the end of the interval right there, 6 minus a, which is the start of the interval, or 0, all divided by the number of rectangles, which in this case, we have an infinite number of rectangles. But I can't just put infinity in the bottom of the fraction. It's not really a number, so I actually have to leave this as just n. So So what we end up getting is that delta x is 6 minus 0 over n or 6 over n. So that's our delta x. Now here we have delta x calculated, so I can go ahead and put 6 over n into this summation. But now we need to find f of x k. Now f of x k, that's the height, because remember this is the base, that's the height, but how do I find the height of each of these rectangles? Well, what you first need to do is find what this x of k inside the function is, and x of k is going to be a, the start of your interval, plus k times delta x. Now this might look kind of confusing, but all this is really saying is that if we're using, say, right endpoints to approximate, well, in this case, we would just keep counting up for each rectangle we have. And this cable would be like, well, we have 1 delta x or 2 delta x or 3 delta x. We've already seen that before, so that's all this is really saying. We're just using variables to describe that. So for x of k, what I need to do is I need to take a, which is the start of our interval 0, and then add this to k times delta x, which we figured out was 6 over n. So what we're going to end up having for x of k is that this is all equal to 6 k over n. So that's our x of k. So now that we have delta x and x of k calculated, what I can now do is plug in this function here f of x k, which is going to be f of this value 6 k over n. So now we have everything plugged into our equation, and at this point, I just need to do some algebra to get this figured out. Now this argument right here that you see, going to wanna keep this consistent for every time you are doing a step here. But because I wanna save some space here, I'm just going to say, remember this argument is out in front for each step that we do, because I'm mostly just gonna focus on simplifying this part right here. So we have 6 over n, that's our delta x. And to find f of 6 k over n, well, that means I need to take this argument right here and plug it into our function. Our function is 1 half x, so that means we're going to have 1 half of this value 6 k over n. Now what I can do is take this 6 right here and divide it by this 2, that's going to give us 3. So we're going to have 3 over n, and keep in mind this is all going to be what this area is equal to, so we're going to have 3 over n and that's going to be times 6 k over n. Now 3 times 6, that's going to give us 18. So we're gonna have 18 k, and then n times n is n squared. So this is what everything simplifies to when we go ahead and plug this into the function, and we reduce this down as far as we can get. Now keep in mind this argument we're matching is still out in front here. And what I can actually do at this point is I'm trying to get this into a certain form with my summation, because this summation stays in the argument. And if I look over here at these rules of summation that we've learned about, if I can somehow get just the k that we have right here to be by itself, well, that's going to allow me to get a form of a summation that I can simplify with the n's that I have right here. So what I'm actually going to do is take this 18 over n squared, and I'm going to bring it outside of the summation. So I'm going to have 18 over n squared on the outside, meaning all we're going to have left over is this k on the inside. And this is perfect right here, because remember this summation that we have there, we know this is a familiar form, it's n times n plus 1 over 2. So what that means is that for this step, I'll keep this limit as n approaches infinity, and I'll also keep this 18 over n squared, but this entire thing for the summation of k, that's just going to become n times n plus 1 over 2. So we're going to have n times n+one over 2. And now things are looking good, because we have n approaching infinity, and we only have n's in this argument right now. So as long as I can get this simplified to a form that looks familiar, well, we can solve this problem. So let's keep going. So we're gonna have the limit as n approaches infinity, and what I'm going to do is I'm going to cancel the one of these n's with the n up there. So you're going to have 18, that's going to be times n plus 1, and then 2 times n would just give us 2 n. And what I can actually do is divide this 18 by 2, which is going to give us 9. So we're gonna have limit as n approaches infinity, and then we're gonna have 9 times n plus 1 all over n. Now what I'm going to do is distribute this 9 into the parenthesis. So we're gonna keep this limit as n approaches infinity because we still haven't applied it yet. Then we're going to have 9 n+9 over n. Now at this point, we know there are some shortcuts when dealing with limits that approach infinity. So what I need to do is look for wherever the highest power of n is in this fraction. Well, I noticed the highest power of n on top is n to the first power, the highest power of n on bottom is n to the first power. Since I have equal powers, equal highest powers on top and bottom, then the answer to this limit is going to be a ratio of the coefficients out in front, which is gonna be 9 over 1. So our area is going to be 9 over 1, which is just 9, and that right there is the area underneath this curve and the solution to this problem. So that is how you can find the exact area underneath the curve of a function using the limit definition. So that is the solution, and this is how you can solve these types of problems. So hope you found this video helpful. Let me know if you have any questions, and let's move on.

Find the area under the curve, of the function $f\left(x\right)=3x+2$ from $x=0$ to $x=4$ using limits.

• A. 32
• B. 56
• C. 26
• D. 50

Find the area under the curve, of the function $f\left(x\right)=x^2$ from $x=0$ to $x=8$ using limits.

• A. 256
• B. $512$
• C. $\frac{1}{3}$
• D. $\frac{512}{3}$