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
Our Greatest Delusion
I’m not sure what I expected to find when I went to Chernobyl. I mean, it’s been so long since the nuclear reactor there melted down and spewed radioactive atoms across the land. So for almost thirty years, this place has been virtually abandoned. These d…
This Black Hole Could be Bigger Than The Universe
We proudly present to you: The kurzgesagt Guide to Curiosity. Join us on an interactive adventure across 160 thrilling pages that will change your perspective on the world forever. Available now on the kurzgesagt shop. You might be inside a black hole th…
Bill Ackman: How to Get RICH During Inflation (RARE New Interview)
Again, my view is inflation, or kind of the house view, is inflation is going to be persistently higher. That can happen in the very short term, like literally weeks. I think the structural forces have changed. Billionaire investor Bill Amman just issued …
Example dividing a whole by a unit fraction
Let’s think about what 3 divided by 1⁄4 is equal to. Pause this video and see if you can figure it out on your own. And I’ll give you a hint: take three holes and divide it into pieces, or sections, that are each one-fourth of a hole. Then think about how…
Identifying proportional relationships from graphs | 7th grade | Khan Academy
We are asked how many proportional relationships are shown in the coordinate plane below, and we have the choices. But let’s actually look at the coordinate plane below to think about how many proportional relationships are depicted here. So pause this vi…
Discussing Reincarnation in Hinduism | The Story of God
[Music] We think of Hinduism as having reincarnation and life after life after life. But there is really an end to that. That they’re hoping and seeking, which is to go to enlightenment. So once they’ve gone through enough, and I guess learned enough, th…