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

Definite integral properties (no graph): breaking interval | AP Calculus AB | Khan Academy


3m read
·Nov 11, 2024

We're given that the definite integral from one to four of f of x dx is equal to six, and the definite integral from one to seven of f of x dx is equal to eleven. We want to figure out the definite integral from four to seven of f of x dx.

So, at least in my brain, I'm visualizing these as areas between the curve y equals f of x and the x-axis. Let's just draw that. We don't know exactly what f of x is, but we can draw an arbitrary f of x just to help us visualize things.

Let me draw. So, if that is, draw that in a bolder color. That's our y-axis, and this is our x-axis. Let's see, all the action is happening between x from 1 to 7.

We could go one, let me we can go one, two, three, four, five, six, seven, and we can even go to 8 if we like. But the important numbers, let's see, we're dealing with 1, 2, 3, 4, 5, 6, 7, and then we go to 8.

Let me just draw the graph y equals f of x, and I'm just going to draw something arbitrary here. So, let's say the graph of y equals f of x looks like that. y is equal to f of x, of course.

Let me label my axes. That's the x-axis; that is the y-axis. Now let's think about what each of these integrals represent. The integral, the definite integral from 1 to 4, well, that's going to be—we're going to be going from 1 to 4 right over here.

So, this is the definite integral from 1 to 4, this area under the curve between the curve and the x-axis dx, which is equal to 6. Now let's see, we also have the region that goes from four to seven. We have this region right over here, and that area is represented by this definite integral—the one that we need to figure out, the definite integral from four to 7 of f of x dx.

We need to figure that out. And they also—what else do we have? So let me underline this. So, that's the area of the region between x equals 4 and x equals 7, under y equals f of x, above the x-axis.

Then they also gave us this last piece of information, which is—let me do this in another color—the definite integral from 1 to 7. Well, that's going from 1 all the way to 7.

So, that's the sum of these two regions right over here. We could rewrite this as the definite integral from 1 to 4 of f of x dx plus the definite integral from 4 to 7 of f of x dx is equal to the definite integral from 1 to 7 of f of x dx.

Notice what's going on here: this first one just goes the area from one to four. Then we go from four to seven. So if you add those together, that's going to be the area from 1 to 7.

They give us a lot of this information. They tell us that this right over here is 6. Let me do that same color. They tell us that this is 6; they tell us that this is 11. So we have 6 plus this is equal to 11.

Well, 6 plus what is equal to 11? Well, this thing right over here must be equal to this thing, right? The definite integral from four to seven must be equal to 5. This must be equal to 5.

Another way to think about it, if this region right over—if I'm having trouble switching colors—if this region right over here is 6, so that has an area of 6, and the whole region, if everything, has an area of 11.

So if that plus that has an area of 11, then the stuff that we don't know, this orange region, this orange region is going to be 11 minus 6. So, this region right over here is going to have an area of 5.

More Articles

View All
You Won't Get Rich Renting Out Your Time
Next, you go into more specific details on how you can actually get rich and how you can’t get rich. The first point was about how you’re not going to get rich. You’re not going to get rich renting off your time. You must own equity, a piece of a business…
Evolutionarily Stable Strategies ft. Richard Dawkins
What are evolutionarily stable strategies? In many cases, it is kind of clear what is the best thing for an individual to do, the best thing it can do to increase its survival with deduction. But there are times when what is the best thing to do depends …
Watch Adorable Babies Go on a Hilarious High-Altitude Adventure | Short Film Showcase
Shelby was doing stuff that no one else was even trying, and a lot of people didn’t even realize he was a baby in the late 2016’s. Like everything had already been done, you know? At that point, the scene shifted to the sub six Monon old group. Tons of ta…
Vitalik: Ethereum, Part 1
All right, welcome everybody back to the podcast. We have with us Haseeb Qureshi, who’s our partner at Dragonfly. Haseeb and I used to work together back when I was more active in crypto land. Vitalik is, of course, a polymath ingenue, although he may bri…
This one thing happens when you own a private jet...
Whatever you think you’re going to use when you own a plane, you will do it 50 to 80% more when you have that airplane. Cuz it’s just the ease of just calling up and saying, “I want to go tomorrow to such and such a place, and we’ll come back tomorrow ni…
15 Habits You Need To Stop Doing Immediately
Here’s a hard pill to swallow: we’ve all got 24 hours in a day. So how is it that some manage to find success while others struggle in despair? Well, it all boils down to one thing: habits. Make no mistake, what you do on a daily basis will dictate your d…