Knowledge Node - Integral Calculus

The inverse function to the derivative is the intergral. Instead of finding the slope, the integral finds the area under a function. The idea behind it is similar to the derivative, using infinitely small parts to find a solution.

Suppose we wanted to find the area under this curve:

One idea is to break it into a bunch of small rectangles. We know the formula to calculate an area of a rectangle, so we can add up all the areas to get an approximate area for the total area of the function. As you'll notice, rectangles don't fit functions very well, but it's a step in the right direction!

The formula to add up the areas of 10 rectangles is below. Note that the formula actually adds up rectangles placed in a different fashion (rather than the middles intersecting with the curve, the right sides of each rectangle intersects with the curve, this difference matters, but not for explanation purposes).

Not too bad, but as we learned in Differential Calculus, smaller parts are always better! Instead of using 10 rectangles, let's try using 100!

Notice that the error is a lot smaller when we use 100 rectangles than when we use just 10. In fact, it looks almost like a perfect match! The formula to add up all the rectangles is below.

Note that our answer is now much more accurate. To be perfectly accurate, an infinite number of infinitely small rectangles must be used. This is described using the integral. Taking the integral of f(x) yields F(x). The integral is the curvy s shape.

When evaluating this integral, the final answer of the area is exactly:

That was pretty easy. Now say we wanted to calculate the area under between these two curves (one is the sin(x) and the other is f(x)=x^2 - 4. They are pictured below:

First we need to find where the two graphs intersect. This is accomplished by setting the two graphs equal to each other and solving for the x-coordinates.

Now that we know their intersection points, we can draw the two curves with rectangles overlaying them.

Once again, we want to use infinitely small rectangles. This is accomplished by taking the limit on the number of rectangles at infinity.

We can describe this situation by taking the area under the sin(x) curve subtracted from the area under the 4-x^2 curve. In integral form, looks something like this:

Integrals can be said to add dimensions to graphs. The formulas created by an n-dimensional integral is n+1 dimensions. Likewise, derivatives take an n-dimensional formula and make an n-1 dimensional formula. That's why the derivative of 2-dimensional n^2 is a 1-dimensional 2n, and the integral of 2-dimensional n^2 is a 3-dimensional graph, n^3/3.

An application of integrals beyond finding areas under curves, is finding volumes of functions rotated around an axis. For example, imagine the following graph is rotated around the x-axis, making a 3-dimensional object. What is the volume of this object?

First, we know that the area of a circle is:

Instead of integrating an infinite number of thin lines to make an area, we are going to integrate an infinite number of thin circles to find a volume. Instead of having a constant r value, we want a variable. This radius is the height of our function, or just plane variable f.

When we substitute in cos(x) for f, we have:

This integral evaluates to:

So the function f = cos(x) when revolved around the x-axis and integrated from 0 to 1, has a total volume of about 2.285 cubic units. This is called the disc (or disk) method.

Integral Calculus is another vast field of research. I highly recommend the first reference link below, as it has an extensive library of information regarding integrals. To evaluate any integral try using the Integrator.