Properties of definite integrals
Worked example: Merging definite integrals over adjacent intervals
- [Instructor] What we have here is a graph of y is equal to f of x, and these numbers are the areas of these shaded regions, these regions between our curve and the x-axis. What we're going to do in this video is do some examples of evaluating definite integrals using this information and some knowledge of definite integral properties. So let's start with an example. Let's say we want to evaluate the definite integral going from negative four to negative two of f of x d of x plus the definite integral going from negative two to zero of f of x dx. Pause this video, and see if you can evaluate this entire expression. So this first part of our expression, the definite integral from negative four to negative two of f of x dx, we're going from x equals negative four to x equals negative two. And so this would evaluate as this area between our curve and our x-axis, but it would be the negative of that area because our curve is below the x-axis. And we could try to estimate it based on the information they've given us, but they haven't given us exactly that value. But we also need to figure out this right over here. And here we're going from x equals negative two to zero of f of x d of x, so that's going to be this area. So if you're looking at the sum of these two definite integrals, and notice the upper bound here is the lower bound here, you're really thinking about, this is really going to be the same thing as, this is equal to the definite integral going from x equals negative four all the way to x equals zero of f of x dx. And this is indeed one of our integration properties. If our upper bound here is the same as our lower bound here and we are integrating the same thing, well, then you can merge these two definite integrals in this way. And this is just going to be this entire area. But because we are below the x-axis and above our curve here, it would be the negative of that area. So this is going to be equal to negative seven. Let's do another example. Let's say someone were to ask you, walk up you on the street and say, quick, here's a graph, what is the value of the expression that I'm about to write down, the definite integral going from zero to four of f of x dx plus the definite integral going from four to six of f of x dx? Pause this video, and see if you can figure that out. Well, once again, this first part right over here, going from zero to four, so what would be is it would be this area. It would be this five right over here, but then we would need to subtract this area because this area is below our x-axis and above our curve. We don't know exactly what this is. But luckily we also need, we need to take the sum of everything I just showed, but plus, plus this right over here. And this, we're going from four to six, so it's going to be this area. Well, once again, when you look at it this way, you can see that this expression is going to be equivalent to taking the definite integral all the way from zero to six, zero to six of f of x dx. And once again, even if you didn't see the graph, you would know that because, in both cases, you're getting the definite integral of f of x dx. And our upper bound here is our lower bound here. So once again, we're able to merge the integrals. And what is this going to be equal to? Well, we have this area here, which is five. And then we have this area, which is six. That was given to us. But since it's below the x-axis and above our curve, when we evaluate it as a definite integral, it would evaluate as a negative six. So this is going to be five plus negative six, which is equal to negative one. And we're done.