2.1 Introduction to integrals

From Förberedande kurs i matematik 2

Current revision

Area under the curve of a function

We have previously found that the gradient of a curve of a function is interesting: it gives us information about how the function changes and has great significance in many applications. In a similar way the area between the curve of a function and the x-axis is of importance. Of course, it is dependent on the curve's appearance and thus closely related to the function in question. It is easy to see that this area has practical significance in many different contexts.

If an object is moving, we can illustrate its speed v plotted against time t in a v,t-diagram. We can see in the figure below three different hypothetical examples:

The distance travelled is in each case

 s(6) = 4\times 3 + 6\times 3 = 30\,\mbox{m},\quad
 s(6) = \frac{6\times 6}{2} = 18\,\mbox{m}\,\mbox{.}

In each cases, you see that the distance travelled by the object is matched by the area under the curve.

More examples of what the area under a curve can symbolise are shown below.

The notation for the definite integral.

In order to describe the area under the curve of a function in symbolic form one introduces the integral sign \displaystyle \,\smallint\, :

 = \int_{a}^{\,c} f(x)\, dx\,\mbox{.}

The antiderivative and the indefinite integral

The function \displaystyle F is an antiderivative (or primitive function) for \displaystyle f if \displaystyle F'(x) = f(x) in any interval. If \displaystyle F(x) is an antiderivative for \displaystyle f(x), it is clear that \displaystyle F(x) + C is as well, for any constant \displaystyle C. In addition, it can be shown that \displaystyle F(x) + C gives all possible antiderivatives of \displaystyle f(x); the expression \displaystyle F(x) + C is known as the indefinite integral of \displaystyle f, and is written

The relationship between definite and indefinite integrals

We have previously found that the area under the curve of a function, i.e. its definite integral, is dependent on the form of the curve. It turns out that this dependence makes use of the antiderivative, which also allows us, in certain circumstances, to calculate such an area exactly.

Suppose that \displaystyle f is a continuous function in an interval. The value of the integral \displaystyle \ \int_{a}^{b} f(x) \, dx\ then is dependent on the limits of integration \displaystyle a and \displaystyle b, but if one lets \displaystyle a have a fixed value and \displaystyle x be the upper limit, the integral will depend only on the upper limit. To clarify this, we prefer to use \displaystyle t as the variable of integration:

We shall now show that \displaystyle A is in fact an antiderivative of \displaystyle f.

The total area under the curve from \displaystyle t=a to \displaystyle t=x+h can be written as \displaystyle A(x+h) and is approximately equal to the area up to \displaystyle t=x plus the area of the column between \displaystyle t=x and \displaystyle t=x+h, i.e. .

where \displaystyle c is a number between \displaystyle x and \displaystyle x+h. This expression can be rewriten as

If we let \displaystyle h \rightarrow 0 the the left-hand side tends towards \displaystyle A'(x) and the right-hand side tends towards \displaystyle f(x) , i.e. .

Thus the function \displaystyle A(x) is an antiderivative of \displaystyle f(x).

Evaluating integrals

In order to use antiderivatives to calculate a definite integral, we note first that if \displaystyle F is an antiderivative of \displaystyle f then

where the constant \displaystyle C must be chosen so that the right-hand side is zero when \displaystyle b=a, i.e.

which gives that \displaystyle C=-F(a). If we summarise, we have that

 = F(b) - F(a)\,\mbox{.}

We can, of course, just as easily, choose \displaystyle x as the variable of integration and write

 = F(b) - F(a)\,\mbox{.}

Evaluating a definite integral is performed in two steps. First one determines an antiderivative, and then inserts the limits of integration. The usual way of writing this is as follows,

 = \Bigl[\,F(x)\,\Bigr]_{a}^{b} = F(b) - F(a)\,\mbox{.}

Antidifferentiation

To differentiate common functions is not an insurmountable problem: there are general methods for doing this. To perform the reverse operation - that is, find an antiderivative (or an indefinite integral) for a given function - is much more difficult, however, and in some cases impossible! There is no systematic method that works everywhere, but by exploiting the usual rules of differentiation "in the opposite direction" and also by learning a number of special techniques and tricks one can tackle a large number of the functions that turn up.

The usual rules of differentiation give

Compensating for the ”inner derivative”

When differentiating a composite function one makes use of the chain rule, which means that one must multiply by the inner derivative. If the inner function is linear, then the inner derivative is a constant. Thus when integrating such a composite function, one must divide by the inner derivative as a sort of compensation.

Note that this way to compensate for the inner derivative only works if the inner derivative is a constant.

Rules for evaluating integrals

Using the way integration has been defined here, it is easy to show the following properties of integration:

1. \displaystyle \int_{b}^{\,a} f(x) \, dx = - \int_{a}^{\,b} f(x) \, dx\,\mbox{,}\vphantom{\Biggl(}
2. \displaystyle \int_{a}^{\,b} f(x) \, dx + \int_{a}^{\,b} g(x) \, dx = \int_{a}^{\,b} (f(x) + g(x)) \, dx\,\mbox{,}\vphantom{\Biggl(}
3. \displaystyle \int_{a}^{\,b} k \, f(x)\, dx = k \int_{a}^{\,b} f(x)\, dx\,\mbox{,}\vphantom{\Biggl(}
4. \displaystyle \int_{a}^{\,b} f(x) \, dx + \int_{b}^{\,c} f(x)\, dx = \int_{a}^{\,c} f(x)\, dx\,\mbox{.}

Moreover, areas below the x-axis are subtracted, that is, if the curve of the function lies below the x-axis in a region, the integral has a negative value in this region:

The total area is \displaystyle \ A_1 + A_2 = \int_{a}^{\,b} f(x)\, dx - \int_{b}^{\,c} f(x)\, dx\,.

Note . The value of a definite integral can be negative, while an area always has a positive value.

Area between curves

If \displaystyle f(x) \ge g(x) in an interval \displaystyle a\le x\le b then the area between the curves is given by

 - \int_{a}^{b} g(x) \, dx\,\mbox{,}

which can be simplified to

Note that it does not matter whether \displaystyle f(x) < 0 or \displaystyle g(x) < 0 as long as \displaystyle f(x) \ge g(x). The value of the area between the curves is independent of whether the curves are above or below the x-axis, as the following figures illustrate:

 \quad\text{and}\quad A_2 = \int_{b}^{\,c} (2- x) \, dx\,\mbox{.}

 \quad \Leftrightarrow \quad x^2 = \frac{1}{2}
 \quad \Leftrightarrow \quad x = \pm \frac{1}{\sqrt{2}}\,\mbox{.}

 \quad \Leftrightarrow \quad x^3 = 1
 \quad \Leftrightarrow \quad x=1\,\mbox{.}