4- Integration PDF

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Practice about Integration (ENGLISH)...

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Practice 4: Integration 4.1

Symbolic integration with Octave

Throughout this practice we will assume the library symbolic is loaded.

4.1.1

Calculus of primitives

In Octave we can compute primitives using the function int whose syntax is: int(f,x)

where f is a symbolic expression that defines the integrand and x is the integration variable (that we have previously defined). For example, for computing a primitive of f (x) = x2 , we write: int(x^2,x)

4.1.2

Cálculo de integrales definidas

La orden para calcular la integral definida de una función f en un intervalo [a, b], int(f,x,a,b)

Therefore, for computing int(x^2, x, 3, 5)

Z

Z

b

f (x) dx, es a

5

x2 we write: 3

or also, syms x; int(x^2, x, 3, 5)

in case of not having previously defined the variable.

4.1.3

Calculation of improper integrals

In the command int, the ends of the integration interval can be +∞ or −∞, so that it is possible to calculate some improper integrals.

4.2

Approximation of integrals through numerical integration

It is not always possible to calculate exact integrals of the type

Z

b

f (x) dx. For example, it may happen that we do a

not know the expression of the function and that we only have the values that it takes in a set of points. It may also happen that the primitive of the function is not expressed in terms of elementary functions or that the evaluation of the primitive is too expensive. Z b In these cases, we can approximate the value of f (x) dx using some method of numerical integration. a

1

2

Practice 4

4.2.1 trapz command The function trapz allows to approximate following syntax:

Z

b

f (x) dx by means of the compound trapezoid formula and has the a

trapz(X,Y)

where X e Y are row vectors of the same dimension. The vector X = (X(1), ..., X(n)) determines a partition partition of [a, b] and Y(i) = f (X(i)), where i = 1, ..., n.

Exercises Exercise 4.1 Calculate, by hand and using the computer: Z Z Z u2 a) x ln(x) dx b) arctan(x) dx c) √ du u3 + 1

d)

Z

1

2

 2 1 2 x + √ dx 3 x

e)

Z

3

−3

p

9 − x2 dx

Exercise 4.2 Calculate, if possible, the following improper integrals: Z +∞ Z 9 Z +∞ ln x 1 dx p dt c) dx b) a) 2 2 1 + t x |x| 1 −4 −∞

Check if the results are correct using the command limit. Z b Exercise 4.3 Create a function file that approximates the defined integral f (x) dx using the mid-point rule. Use it for finding a Z π 2 √ an approximation of cos x dx. −π 2

Z Exercise 4.4 Approximate, using the compound trapezoid formula,

points, for the cases n = 1, 2, 3, 4, 5.

π 2

√ cos x dx. Use partitions of the interval [0, π] with 10n

− 2π

Exercise 4.5 A corral that has a square shape of 100 meters in length on each side, is fenced. The end of a length rope of length ℓ is fixed to one of its vertices. At the free end of the rope, a cow is tied. 1. Prove that the area (in m2 ) of the surface located inside the corral, where the cow can pasture, is:  π  ℓ2 if 0 ≤ ℓ ≤ 100    4     !  √   √ √ ℓ2 ℓ2 100 ℓ2 − 1002 2 2 A(ℓ) = − 100 ℓ − 100 + if 100 < ℓ < 100 2 arcsen arcsen   ℓ ℓ 2 2       √  104 if ℓ ≥ 100 2

2. Is the function A continuous in its definition domain or not? Make the necessary calculations to justify your answer. 3. Find, using the dichotomy algorithm, an approximation of the value of ℓ for which the cow accesses a surface area equal to half the area of the corral. Use four iterations. Find a bound of the error. 4. Prove that the exact value of the surface that the cow can graze when ℓ = 125 is: A(125) = 7500 +

Z

100 75

p

5. Approximate that value using a numerical integration formula.

1252 − x2 dx...