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© Copyright 2003, Jim Loy
An
integral is the area under a curve. We often describe it as the sum of the tall
thin rectangles as shown on the left. The integral is the Limit of this area as delta x approaches 0. Rectangles
that go below the x axis have negative area. The picture on the left shows the
integral from x = a to b. The notation is defined above right (see
Infinite Series). The integral portion, on
the left of the equal sign, is said as "The integral from a to b of f(x) with
respect to x."
The integral is often extremely useful and important. We can use it to find areas and volumes, which are often important in sciences and engineering.
The integral in the definition above is a definite integral, one in which the endpoints are specified, and its value is a area (which might blow up to infinity, like many sums). There are also indefinite integrals, which have no endpoints. This represents a sum which we cannot evaluate yet, because we don't know the endpoints. But we may know how it behaves.
The integral is the inverse function of the Derivative. And so, we can easily find the integrals of many functions. The derivative of the sine is the cosine, and so the indefinite integral of the cosine is the sine. And the integral of sin(x) is -cos(x). The indefinite integral is a function of x. It has a value for each value of x. And we can use the values of the function at the two end points to find the value of the definite integral.
So how do we evaluate the definite integral of sin(x) from 0 to pi radians? With an integral from a to b, we evaluate the integral at b, and subtract the integral at a. So the value of the integral of this sine function from 0 to pi is: -cos(pi)+cos(0) = 1. So the area under the curve of the sine curve (from 0 to pi) is 1.