# Introduction to Integrals

An integral is useful for finding the area underneath a function. Let $$f(x)$$ be any arbitrary function such that it is smooth and continuous at every point. To find the area underneath $$f(x)$$, we must go through several steps. First, we'll start off by drawing an $$n$$ (where $$n$$ is any positive integer) number of rectangles of equal width underneath $$f(x)$$ as illustrated in Figure 1. What is the total area of all the rectangles? The area of the first rectangle is $$A_1=f(x_1)(x_2-x_1)$$; the area of the second rectangle is $$A_2=f(x_2)(x_3-x_2)$$; and the area of the $$n$$th rectangle is $$A_n=f(x_n)(x_{n+1}-x_n)$$.  Since every rectangle has the same width, it follows that $$x_2-x_2=x_3-x_2=x_{n+1}x_n= Δx$$. To find the total area of all the rectangles, let's add up the area of each rectangle:

Figure 1

$$A=A_1+A_2+...+A_n=f(x_1)Δx+ f(x_2)Δx+...+ f(x_n)Δx\sum_{i=1}^nf(x_n)Δx.\tag{1}$$

As you can see visually in the animation in Figure 2, as the number of rectangles $$n$$ increases, the area $$A$$ becomes closer and closer to equaling the exact area underneath the curve. Using Equations (1), let's take the limit as $$n→∞$$ to get

$$\lim_{n→∞}\sum_{i=1}^nf(x_n)Δx.\tag{2}$$

Figure 2. The approximate area, $$\sum_{i=1}^nf(x_n)Δx$$, underneath the curve $$y=x^2$$ becomes closer and closer to the exact area underneath the curve as the number $$n$$ of terms increases.

Let's review the notion of a limit that we covered in an earlier lesson. The value that the limit, $$\lim_{z→c}g(x)$$, is equal to is the value that $$g(x)$$ gets closer and closer to while $$z→c$$. Take for example the limit, $$\lim_{x→2}x^2$$, that we looked at in a previous lesson. The value that this limit is equal to is the value that $$x^2$$ gets closer and closer to as $$x→2$$.  We showed that this value is $$4$$. Similarly the value of the limit, $$\lim_{n→∞}\sum_{i=1}^nf(x_n)Δx$$, is the value that $$\sum_{i=1}^nf(x_n)Δx$$ gets closer and closer to equaling the exact area underneath the curve $$f(x)$$. Thus, the limit must equal the area underneath $$f(x)$$ and

$$\text{Area underneath f(x)}=\lim_{n→∞}\sum_{i=1}^nf(x_n)Δx.\tag{3}$$

Let's see if there is a simpler way of rewriting the right-hand side of Equation (3). Not to sound too annoyingly repetitive but, again, the limit $$\lim_{n→∞}\sum_{i=1}^nf(x_n)Δx$$ is equal to the thing that $$\sum_{i=1}^nf(x_n)Δx$$ gets closer and closer to equaling as $$n→∞$$. But if you think about it for a moment, the following must be true: if $$n→∞$$, then the number $$n$$ of the terms $$f(x_i) Δx$$ is getting closer and closer to infinity; thus, the finite sum $$\sum_{i=1}^n$$ (of an $$n$$ number of terms) is getting closer and closeer to becoming an infinite sum. Let's represent an infinite sum (that's to say, a sum of infinitely many terms) by the symbol "$$∫$$." As $$n→∞$$, it is also true that the width $$Δx$$ is getting closer and closer to becoming infinitely small. Let's represent an infinitely small $$Δx$$ by the symbol " $$dx$$ ."

With all that said, I'd like to just make a few remarks about the variable $$x$$, then about the expression $$\int{f(x)dx}$$, and then we'll see how that ties in with our discussion of the limit $$\lim_{n→∞}\sum_{i=1}^nf(x_n)Δx$$. Since $$x$$ is a continuous variable, it can take on an infinite number of values: such as the numbers $$2$$, $$π$$, $$3.001$$, $$3.00001$$, $$3.00000001$$, etc. Thus, there are an infinite number of y-values along the curve $$f(x)$$: including $$f(2)$$, $$f(π)$$, $$f(3.001$$, etc. The term $$f(x)dx$$ is the area of an infinitely skinny rectangle; and the expression $$\int{f(x)dx}$$ is the sum of an infinite number of the terms, $$f(x)dx$$. $$\int{f(x)dx}$$ gives the infinite sum of all the areas $$f(x)dx$$ of (infinitely) skinny rectangle. What is $$\sum_{i=1}^nf(x_i)Δx$$ getting closer and closer to equaling as $$n→∞$$? Well, clearly its getting closer and closer to being an infinite sum, and $$x_i$$ and $$Δx$$ are approaching $$x$$ and $$dx$$. Thus, the limit $$\lim_{n→∞}\sum_{i=1}^nf(x_n)Δx$$ must also equal that thing and

$$\int{f(x)dx}=\lim_{n→∞}\sum_{i=1}^nf(x_i) Δx.\tag{4}$$

The expression $$\int{f(x)dx}$$ is called "the integral of $$f(x)$$ with respect to the variable $$x$$" and it is equal to two things: first, the area underneath $$f(x)$$; second, it is also the sum of the infinite number of the terms $$f(x)dx$$ and is the infinite sum of the area of infinitely many, infinitely skinny rectangles. My apologies, the latter is quite a mouthful. But hopefully this lesson helped give you a better idea of what an integral actually is. In the next several lessons, we'll investigate techniques for solving integrals - that is, finding the area underneath various different functions $$f(x)$$.