Integration is a method that can be used to calculate lengths, areas, and volumes defined by mathematical functions. It also has many applications in pure mathematics, physics, statistics, and many other fields. According to the Fundamental theorem of calculus integration and differentiation are (loosely speaking) inverse processes.
This article is an overview of integration. We will look at some of the key concepts but as this is just an overview we will not prove them or explain them in great detail.
Integration as area under the curve
A simple definition of the integral of a function is that it is the area under the curve of that function. Here, on the left, is a simple function, y = 1. The graph on the right shows the area under the curve at every point x:
The shaded area in the left-hand graph shows that the area under the curve between 0 and 2 is a rectangle. The rectangle has width 2 and height 1 so its area is 2. The point marked on the right-hand graph shows the value of the integral at that point, which is indeed 2.
In general, the area under the curve between 0 and x is an x by 1 rectangle, so its area is x. The graph on the right, of course, is y = x.
In this next example, the curve on the left is y = 2x:
The shaded area under the curve on the left is a right-angled triangle with width 2 and height 4, so its area (half base times height) is 4. The point marked on the right-hand graph shows the value of the integral at that point, which again matches the value 4.
How can we find the formula of the curve on the right? In general at the point x, the width of the triangular area will of course be x, and the height will be 2x (from the formula of the line). So the area will be:
In these simple examples, it was easy to calculate the area using simple geometry. Most cases aren’t quite so simple. For…