yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Functions defined by integrals: switched interval | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

The graph of f is shown below. Let G of X be equal to the definite integral from 0 to X of f of T DT.

Now, at first when you see this, you're like, "Wow, this is strange! I have a function that is being defined by an integral, a definite integral, but one of its bounds is X." You should just say, "Well, this is okay. A function can be defined any which way," and as we'll see, it's actually quite straightforward to evaluate this.

So, G of -2. G of -2, and I'll do the -2 in a different color. G of -2, well, what we do is we take this expression right over here, this definite integral, and everywhere we see an X, we replace it with a -2. So, this is going to be equal to the integral from 0 to X of, and I'll write X in a second, F of T DT. Well, X is now -2, this is now -2.

Now, how do we figure out what this is? Now, before we even look at this graph, you might say, "Okay, this is the region under the area, the region under the graph of F of T between -2 and 0." But you have to be careful. Notice our upper bound here is actually a lower number than our lower bound right over here.

So, it will be nice to swap those bounds so we can truly view it as the area of the region under F of T above the T-axis between those two bounds. When you swap the bounds, this is going to be equal to negative definite integral from -2 to zero of F of T DT.

And now what we have right over here, what I'm squaring off in magenta, this is the area under the curve F between -2 and 0. So, between -2 and zero, that is this area right over here that we care about.

Now, what is that going to be? Well, you could—there's a bunch of different ways that you could do this. You could split it off into a square and a triangle. The area of this square right over here is four; it's 2 by 2.

Just make sure to look at the unit; sometimes each square doesn't represent one square unit, but in this case it does, so that's four. Then up here, this is half of four, right? If it was all of this, that would be four. This triangle is half of four, so this is two right over there.

Or you could view this as base times height times 1/2, which is going to be 2 times 2 times 1/2. So, this area right over here is six. So, this part is six, but we can't forget that negative sign, so this is going to be equal to negative six. Thus, G of -2 is -6.

More Articles

View All
Space Mountain Fears - Smarter Every Day 12
Intro music Hey. It’s Disney World, and it’s magic hours, which means nobody is here, so we get to ride everything. But we’ve always had this fear of Space Mountain; that if you put your arms up, you’ll get ‘em chopped off. So we got this trick we do, sh…
Quit Seeking APPROVAL Like a Loser (Stoic SELF-RESPECT) | STOICISM
How often have you felt like you were going through the motions, living your life based on the expectations of others, like you’re an actor continuously seeking approval and validation from an invisible audience? If so, you’re not alone. Like you, million…
Why I’ll never use Stash investing
What’s the guys? It’s Graham here. So, after posting my review on Acorns Investing, many of you have asked that I review another investing app known as Stache. And no joke, this was such a popular request! At least a few hundred of you have asked for this…
Example translating points
What we’re going to do in this video is look at all of the ways of describing how to translate a point and then to actually translate that point on our coordinate plane. So, for example, they say plot the image of point P under a translation by five unit…
Infinity
So imagine you’re Usain Bolt. You’re like six and a half feet tall. You have a couple Olympic medals—no biggie. You show up to your final Olympic race, and your only opponent is a tortoise. For some reason, he has gold medals around his neck. He can talk …
Here's how to save $10,000 in 6 months.
[Music] Hey guys, welcome back to the channel. In this video, we’re going to be talking about how to save $10,000 in six months. This is a goal that I have set myself in the past; I’ve actually been able to achieve it quite a few times. So, in this video…