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
How to Brute Force your way to $1 Million
Let’s get something out of the way: one million dollars isn’t what it used to be. Yes, it won’t be enough to live a lavish lifestyle for the rest of your life, but it would definitely make your life exponentially better than it is right now. Here’s someth…
Mr. Freeman, part 08 — "Оскорблять меня весело и безопасно!"
white noise Hello there! My name is Mr. Freeman and I have a small mental dick. I’m black & white, like bird shit, because it’s cheaper! I’m screaming like a tantrum. I’m Captain Obvious with di… …Napoleon complex! I am scum and noobs cheating! I’m mi…
Great Schism or East-West Schism part 1 | World History | Khan Academy
In previous videos, we talked about the dramatic turnaround in the 4th century in terms of how Christianity was treated in the Roman Empire. As you enter into the 4th century, it’s persecuted by Diocletian, but then Constantine takes power. He’s sympathet…
Artificial selection and domestication | Natural selection | AP Biology | Khan Academy
Most of us are familiar with dogs, oftentimes known as man’s best friend. What’s fascinating about them is that they are one species, even though different types of dogs, different breeds, could look very, very different. The fact that they’re one species…
Peter Lynch: How to Turn $10,000 Into $100,000 in the Stock Market
The goal of this video is to help you find stocks that have 10x return potential. One of my favorite investors of all time, Peter Lynch, calls these type of stocks “10 baggers.” These are the type of investments that pay off so well that they make investo…
5 realistic side hustles for an extra $500-$1000/month
If you’re looking for a realistic video about some side hustle ideas without the purpose of selling you courses on how to get rich, you’re in the right place. You see, because most of the videos you see online talk about various side hustle options and th…