Click here to purchase the entire book in PDF format. Integration and Area If I asked you to calculate the area of a rectangle with the dimensions 2 cm x 3 cm, you'd probably already know that you calculate this by multiplying the length by the width. Therefore 2 cm x 3 cm = 6 cm$^{2}$ or 6 square centimeters. If you were asked to calculate the area of a triangle, the math would be a little bit more complicated, but not much more. What happens when you have to find the area of a more complicated shape that includes a curved side? That's the problem we're going to look at in this section. Let's look at the graph of the function $y = \sin(x)$ from $x=0$ to $x=\pi$ as is shown in Figure 1.22. Figure: $y = \sin(x)$ for $0 \leqslant x \leqslant \pi$ We've seen in the previous section how to express the equation for finding the slope of this function, but what if we wanted to find the area under the graph? How do we find this? Well, the way we found the slope was initially to break the curve into shorter and shorter straight line components. We'll do basically the same thing here, but we'll make rectangles by creating simplified slices of the area under the curve. Figure: $y = \sin(x)$ for $0 \leqslant x \leqslant \pi$ divided into 10 rectangles For example, compare Figure 1.22 to Figure 1.23. What we've done in the second one is to consider the sine wave as 10 discrete values, and draw a rectangle for each. The width of each rectangle is equal to $\frac{1}{n}$ of the total length of the curve where $n$ is the number of rectangles. For example, in Figure 1.23, there are 10 rectangles (so $n=10$) in a total width of $\pi$ (the length of the curve on the x-axis) so each rectangle is $\frac{\pi}{10}$ wide. The height of each rectangle is the value of the function (in our case, $\sin(x)$ for the series of values of $x$, $\frac{0\pi}{n}$, $\frac{1\pi}{n}$, $\frac{2\pi}{n}$, and so on up to $\frac{n \pi}{n}$. If we add the areas of these 10 rectangles together, we'll get an approximation of the area under the curve. How do we express this mathematically? This is shown in Equation 1.81. $$A_{n} = \sum_{i=0}^{9} \sin \left ( \frac{i}{10} \right ) \frac{\pi}{10}$$ (2.81) What does this mess mean? Well, we already know that $n$ is the number of rectangles. The $A_{n}$ on the left side of the equation means ``the area contained in $n$ rectangles.'' The right side of the equation says that we're going to add up the product of the sin of some changing number and $\frac{\pi}{10}$ ten times (because the number under the $\sum$ is 1 and the number above is 10). Therefore, Equation 1.81 written out the long way is shown in Equation 1.82 $$A_{n} = \sin \left ( \frac{0 \pi}{10} \right ) \frac{\pi}{10... ... + ... + \sin \left ( \frac{9 \pi}{10} \right ) \frac{\pi}{10}$$ (2.82) Each component in this sum of ten components that are added together is the product of the height ( $\sin \left( \frac{i \pi}{10} \right )$) and width ( $\frac{\pi}{10}$) of each rectangle. (All of those 10's in there are there because we have divided the shape into 10 rectangles.) Therefore, each component is the area of a rectangle, which, when all added together give an approximation of the area under the curve. Figure: $y = \sin(x)$ for $0 \leqslant x \leqslant \pi$ divided into 20 rectangles Figure: $y = \sin(x)$ for $0 \leqslant x \leqslant \pi$ divided into 50 rectangles Figure: $y = \sin(x)$ for $0 \leqslant x \leqslant \pi$ divided into 100 rectangles There is a general equation that describes the way we divided up the area called the Riemann Sum. This is the general method of cutting a shape into a number of rectangles to find an approximation of its area. If you want to learn about this, you'll just have to look in a calculus textbook. Using rectangular slices of the area under the curve gives you an approximation of that area. The more rectangles you use and the thinner they are, the better the approximation. As the number of rectangles approaches infinity, the approximation approaches the actual area under the curve. There is a notation that says the same thing as Equation 1.81 but with an infinite number of rectangles. This is shown in Equation 1.83. $$\int_{0}^{\pi} \sin (x) dx$$ (2.83) What does this weird notation mean? The simplified version is that you're adding the areas of infinitely thin rectangles under the curve $\sin(x)$ from $x=0$ to $x=\pi$. The 0 and $\pi$ are indicated below and above the S-shaped $\int$ sign.2.6 On the right side of the equation you'll see a $dx$ which means that's it's the $x$ that's changing from $0$ to $\pi$. This equation is called an integral which is just a fancy word for ``the area under a function.'' Essentially, just like ``derivative'' is another word for ``slope,'' ``integral'' is another word for ``area.'' The general form of an integral is shown in Equation 1.84. $$A = \int_{a}^{b} f(x) dx$$ (2.84) Compare Equation 1.84 to Figure 1.27. What it's saying is that the area $A$ is equal to all of the vertical ``slices'' of the space under the curve $f(x)$ from $a$ to $b$. Each of these vertical slices has a width of $dx$. Essentially, $dx$ has an infinitely small width (in other words, the width is 0) but there are an infinite number of the slices, so the whole thing adds up to a number. Figure 1.27: A graphic representation of what is meant by each of the components in Equation 1.84 [Edwards and Penny, 1999].