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

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.