Transcription
Earlier, to find the area under the graph of this function, we found out the lower sum. That is, in each sub-interval, we took the minimal ‘y’ values as the heights of the rectangles. And we saw that as the number of sub-intervals ‘n’ tends to infinity, the lower sum approaches the area under the graph. But is there some significance behind only considering the minimum value in each sub-interval? Why do we take the heights of the rectangle as the minimum ‘y’ values only in each sub-interval? What if we take the maximum ‘y’ value? Would this work? Yes! It would. Let’s quickly have a look.
Notice that in each sub-interval the value of ‘y’ corresponding to the right end value of ‘x’ will be the maximum ‘y’ value. Taking these as the heights of the rectangles, we will get the sum of their areas to be this. Take a moment to understand this. Now notice that this sum of areas is greater than the area under the graph. Even if we increase the number of sub-intervals, this sum will always be greater. So this sum is called the upper sum. We denote it by putting a bar above the symbol. Now we can simplify this like we did earlier to get this expression. From this, we can see that as ‘n’ tends to infinity, the term ‘8 over n’ will approach 0. So we see that in this case also, the sum of the areas of the rectangles will approach 16. Therefore, we get that the height of the rectangles can be the minimum ‘y’ values in each sub-interval or the maximum “y” values. In the first case, we get the lower sum and in the second we get the upper sum. And in the limit ‘n’ tends to infinity, both the sums approach the area under the graph.
Now there could be another important question! What if we take any ‘y’ value in between the maximum and the minimum in each sub-interval? Will this also work? Yes, it would! Consider one particular sub-interval. Let’s take any ‘y’ value as the height of the rectangle. As this is the third interval, let’s denote its area by ‘A3’. Then the area of this rectangle will be less than the area of this rectangle. And it will be greater than the area of this rectangle. That is, this area will always be between these two areas. This is true for any sub-interval. So the sum of the areas of these types of rectangles will always be between the lower sum and the upper sum. But we know that as ‘n’ tends to infinity, the lower sum and the upper sum approach the area under the graph. So the sum of these areas will also approach the area under the graph. This tells us that whatever ‘y’ value we take as the height of the rectangles, the limit of the sum of areas of rectangles will approach the area under the graph. This limit of the sum of areas of the rectangles is called the Integral of the function. In this case, we will say that the integral of the function between x equal to 2 and 6 is ‘16.’ It’s denoted like this.
In the next part, let’s summarize the whole process of integration and understand what this means. Consider this general function. Let’s say, we want to find the area between x equal to ‘a’ and x equal to ‘b’; that is, the integral of this function between x equal to a and b. We saw that this integral is equal to the limit of the sum of the areas of the rectangle. In particular, we always consider the limit of the lower sum and the upper sum. So what are the lower sum and the upper sum? Right! To find the lower sum and the upper sum, we first divide this interval into ‘n’ sub-intervals. For the lower sum, we consider the minimum value of the function in each sub-interval. And for the upper sum, we consider the maximum value of the function in each sub-interval. The rectangles formed considering these values as the height give us the lower and the upper sums. Look at this figure. The lower sum is the sum of the areas of these rectangles. Symbolically, we can write it like this. Now look again! The upper sum is the areas of these rectangles. Symbolically, we can write this sum like this.
From this we can see the origin of the term ‘f of x times dx’ in this symbol. It denotes the area of a rectangle. But wait, why do we use ‘dx’ here instead of Delta X? This is because we know that the more the number of sub-intervals, the better is the approximation. As we increase the number of sub-intervals, the width of the rectangles becomes very very small. So by ‘dx,’ we mean that the width of the rectangles is infinitesimal. What we are really doing is adding the areas of very very thin rectangles. We see that in the limit the number of sub-intervals ‘n’ tends to infinity, these sums approach the area under the graph. So the integral is not the sum of the areas; it’s the limit of the sums. To signify this, we use the long ‘s’ in this symbol instead of the usual Sigma notation for summation. Interesting, isn’t it?
One last thing. Can you guess what the a and the b written here mean? Correct. It tells us the value of ‘x’ between which the area under the graph is evaluated. So we see that this symbol tells us the whole idea of integration. With this process, we can find the area under the graph of a function. Here’s a function for you! Can you find the integral of this function between x equal to zero and x equal to four? Share your thoughts in the comment section below.
Remember we earlier said that the two processes of integration and differentiation are related to each other? Now that we saw how to integrate a function, can you find this connection? We will see this connection in the upcoming videos. Also, didn’t you find the whole process of integration a bit lengthy and cumbersome? Don’t worry. This isn’t the process we always use; it was just for a better understanding. We will see that this connection leads us to a better way to find the integral of a function. Do subscribe to our channel to stay updated and to keep learning.