The Riemann integral

Let's consider the problem of calculating the area under the grapf of the function y = x² for x ∈[0,1].
We divide up the inteval [0, 1] in n equal subintervals

x_{i} = \frac{i}{n}, x_{i + 1} = \frac{i + 1}{n}, i = 0, \dots, n - 1

calcolo area trapezoide x^2 — **Fig.1** - L'area sottostante la curva è approssimata dalla somma delle aree dei rettangoli.

Approssimiamo l'area sottesa la curva, con la somma delle aree dei rettangoli:

For n ⟶ ∞ we obtain

lim_{n \to \infty} S_{n} = \frac{(n - 1) n (2 n - 1)}{6 n^{3}} = \frac{1}{3}

The area is thus 1/3.

Proviamo a generalizzare questo procedimento a qualsiasi funzione, considerando una funzione continua, f: [a, b] ⟶ ℝ. Effettuiamo la partizione dell'intervallo di definizione, in n intervallini, i cui estremi sono individuati dai punti

a = x₀, x₁, x₂, ..., x_n−1, x_n = b

with

x_i = a + ih, h = (b − a)/h, i = 0, ..., n

inside each of the n intervals [x_i−1, x_i] we pick an arbitrary point ξ ∈ [x_n−1, x_i]. We define the sum of Cauchy-Riemann, as

S_{n} = \sum_{i = 1}^{n} f (ξ_{i}) \cdot (x_{i} - x_{i - 1}) = \frac{b - a}{n} \sum_{i = 1}^{n} f (ξ_{i})

The quantity S_n is known as the partial sum.

Passando al limite per n tendente ad infinito si passa da una somma ad un integrale:

Teorema 4.2. Given a continuos function f: [a, b] ⟶ ℝ, there exists finite the limit of S_n, indipendently by ξ_i, called definite integral of f on [a, b] expressed as:

lim_{n \to \infty} S_{n} = lim_{n \to \infty} \frac{b - a}{n} \sum_{i = 1}^{n} f (ξ) = \int_{a}^{b} f (x) d x

If, as we let the number of partions tend to ∞, the partial sum S_n tends to a limit, then we say the function f is Riemann integrable (integrable, hereafter) on [a, b] and the limit to which the partials sums converge is known as the integral of f over [a,b].

A definition of Area

The Riemann integral permits a precise definition of the geometrical concept of "area" under a curve and not viceversa.

Reconsidering the Cauchy-Riemann sum (6.3). Let f ≥ 0 a continuos function on [a,b]. Over each subinterval [x_i−1, x_i] whose height is the value of f(ξ) with ξ a selected point in the subinterval. The sum S_n is an approximation of the area between the graph of f and the x-axis, in the interval a > x > b. This region is known as trapezoid.

lim_{n ⟶ ∞} S_n = ∫^b_{_a} f(x) dx = area of the trapezoid

When f has no fixed sign, the integral measures the difference of the positive regions (above the x-axis) and the negative regions (below it). Consider for example sin(x):

because of symmetry we have:

\int_{0}^{2 π} s i n (x) d x = 0

Length of a Curve

Another important geometrical concept associated with a curve leads to an integration. This is the lenght of arc.

Let y = f(x) a continuous function and f'(x) its interval in the interval [a, b]. An increment dx along the x-axis corresponds to an increment dy = f' dx along the y-axis. Then we have

d l = \sqrt{(d x)^{2} + (d y)^{2}} = \sqrt{(d x)^{2} + f^{'} (x) (d x)^{2}} = \sqrt{1 + f^{'} (x)^{2}} d x

Summing up along all of the infinitesimal distances along the entire length of the curve:

l = \int_{a}^{b} \sqrt{1 + f^{'} (x)^{2}} d x

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