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
Probability with combinations example: choosing cards | Probability & combinatorics
We’re told that a standard deck of 52 playing cards includes 4 aces, 4 kings, and 44 other cards. Suppose that Luis randomly draws four cards without replacement. What is the probability that Luis gets two aces and two kings in any order? So like always,…
Golf Course Camping | Dirty Rotten Survival
As the boy’s head deeper into suburbia, Johnny needs to find a legal place to make camp before it gets too late. What is this? We think it is… it’s a golf course. What’s your stay here? Obviously, this woods is owned by the golf course. “Look, a fire! Ge…
Substitution and income effects and the Law of Demand | APⓇ Microeconomics | Khan Academy
In other videos, we have already talked about the law of demand, which tells us—and this is probably already somewhat intuitive for you—that if a certain good is currently at a higher price, then the quantity demanded will be quite low. As the price were …
Charlie Munger: How to Make Your First $1 Million (5 Steps)
Charlie Munger is currently a billionaire with an estimated net worth of 2.4 billion dollars as of 2022. However, that wasn’t always the case. While Charlie didn’t grow up poor by any means, he wasn’t lucky enough to be born into a rich and prominent fami…
The Science of Curveballs
[Applause] You pitch that! Hey, how did you do that? That was a hard one because, uh, this ball is a little bit magic. It’s got a bit of string glued to the left side of it to make the ball curve to the left. Why is that? And that’s because the air that’…
The Dark Night of the Soul (Losing Who We Thought We Were)
The endurance of darkness is preparation for great light. John of the Cross. Most of our lives are ongoing pursuits of sensory pleasures. And every time we think that we’ve found lasting fulfillment, it doesn’t take long before we need more gratification…