00:01
Okay, so our goal here is to find the area under the graph of f over the given interval, and we have two different piecewise functions here for us that we would like to answer this for.
00:08
So i have gone ahead and graphed these piecewise functions.
00:14
I graphed the first one here, and you can see that a piecewise function is two different functions that may or may not connect at the same point.
00:20
This one does, but here at this point when x equals three is when it splits between the functions.
00:27
So before x equals 3, it is one function, and after x equals 3, it's a different function.
00:33
So the way we can find the area under these graphs on the given intervals is to go ahead and take the integral up until that splitting point.
00:40
So up until we switch from this function to this function, and then add those two integrals together.
00:46
So we're looking for all of this area under here, and then we'll add all of this area under here.
00:51
And that will get us our area underneath the total piecewise function.
00:55
Okay.
00:56
So let's do that for the first one.
00:57
So for the first one, we have f of x is 2x plus 1.
01:00
For x is less than or equal to 3.
01:02
So i can take the integral starting at 1.
01:06
It says here that we would like to start at 1.
01:08
And i can take the integral of the first piece all the way up to 3.
01:11
That's as long as it's relevant.
01:12
After 3, it no longer is this red function.
01:15
It switches to being this blue function instead.
01:17
So i can only start at 1 and go to 3 for the first function.
01:20
So i'll have 2x plus 1, dx.
01:24
And then i can add on the integral of the second function.
01:27
In our case the blue function.
01:29
So i can say then i'll have the integral starting at three and going until five, which is the end of our interval, oops, five, there we go, of 10 minus x dx...