11th Grade - Integrals

11th Grade lesson

4 - Integrals


Somewhere 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 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.
integration math theory

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 integral symbol reads 'sum' but most of the time we say 'integral', and hence the area under the curve is : area and integration, 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').

integration math theory

This is the theory, let's see now how to calculate an integral. It's very easy.

How to calculate an integral

integration calculation formula

Hence :
integration calculation
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 : derivative formula. If two functions are the same then their integrals are the same, so :
integration math equation

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 :
integration math equation

So if you swap sides :
integration math equation

Inverse the equality :
formule integration bye parts

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 antiderivative calculation. 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 integration bye parts formula and math integration formula. Then integration calculation and integration calculation. 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
area and integral
You have to calculate:
area and integral

>>> Mathematical induction lesson >>>

The integrals

lesson, problems