IntegralsSomewhere on Earth, there is a field between a road and a river. The owner of the field dies and the field has to be divided in 3 equal parts for its legatees. We need to know its area. It is the aim of this chapter.
First filter out all the useless data and plunge the field into a frame.
Then let's seek a function whose graph is the bank of the river. Using the calculator, you can find a close function. There is a technique that enables us to calculate this function if you know some points of the graph, and the more points you'll know, the more precise the graph will be. This technique is not in the 12th grade program. For a river, consider that the function works perfectly. To determine the area under the curve, draw side by side rectangles under it. You know how to calculate the area of a rectangle (length times width), then by adding them all you'll find a number a little smaller than the area you want to find.
Below, let's take thiner rectangles. The area is more precise. In fact the smaller their width will be, the more precise the area will be. In fact the area below the graph is the sum of the areas of an infinite number of rectangles, whose width is infinitely small. The infinitely small width is denoted by dx. It's an infinitesimal variation of x. The heigth of every rectangle is f(x). The sign reads 'sum' but most of the time we say 'integral', and hence the area under the curve is : , that means the sum of the f(x) for x between 0 and 4 multiplied by dx. (it reads 'integral from 0 to 4 of f(x)dx').
This is the theory, let's see now how to calculate an integral. It's very easy.
How to calculate an integral
The result is expressed in area units. If a unit of the frame is 5m, then an area unit (ua) of the graph is 25mē, and then the real area of the field is about 267mē.
Other calculation method:
Sometimes you can't find an antiderivative for the function f. In that case, the integration by parts formula can be useful.
You know the formula : . If two functions are the same then their integrals are the same, so :
Because the antiderivative of the derivative of a function is the function and that the integral of a sum of functions is the sum of the integrals, then :
So if you swap sides :
Inverse the equality :
It is the integration by parts formula. It can be used when you can't calculate the antiderivative of a function you want to integrate. For example, for this application, let's calculate . We don't know how to calculate the antiderivative of xcos(x). But because there is a product, let's do an integration by parts. Suppose and . Then and . Hence :
An integral can be negative, whereas an area is always positive. In fact if you want to calculate the area S of the blue area below
You have to calculate: