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.

In this example, we are given the function \( f(x) = x \), and we are asked to find the area under the curve from \( x = 0 \) to \( x = 5 \). Now I can see 0 is right about here on the x-axis, and 5 is right there. 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 \( \frac{1}{2} \text{ base } \times \text{ height} \). So the area is going to be \( \frac{1}{2} \) 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 \( \frac{1}{2} \times 5 \times 5 \), which is \( \frac{1}{2} \) of 25, and \( \frac{1}{2} \) 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.

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. Example a asks us to find the area underneath this function from \( x = -8 \) to \( x = 4 \). I can see -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, I could try to find a trapezoid or some kind of familiar shape here, but the more simple thing to do is going to be to split this up into two shapes that we're familiar with. Notice if I divide the shape in this fashion, we get two areas, area 1 and area 2. Area 1 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 \( \frac{1}{2} \) base times height, plus the area of a rectangle, which is just base times height.

Looking at this, I can see that the base of our triangle is going to be this distance right here. We go from -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 going to 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 \( x = -6 \), which is right there, to \( x = 8 \), 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 is this actually something that's possible to calculate? Well, it turns out that it is. The statement about being under the curve 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. Let's see if we can understand why this area becomes negative. We have a rectangle that forms here, and the area of a rectangle is base times height. Now what I first need to do is calculate the base of this rectangle. The base is going to be this distance, which goes from -6 to 0, so that's a distance of 6, plus this distance, which goes from 0 to 8. \( 6 + 8 = 14 \), 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 -6, that means the height is going to be -6. So we have \( 14 \times (-6) \), which gives us an area of -84. 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.