No discussion of calculus would be complete without
integral calculus even though one could do IAC just fine without ever learning that the integral is the reverse operation of a derivative, up to an additive constant. Integrals of multivariate functions, in her mind, were taught almost entirely in the context of
definite integrals:
indefinite integrals of multivariate functions depend on the order of the integration even though partial differential equations are also about indefinite integrals of multivariate functions. Jessica knew as much about mathematics, and prospective
mechu-deru users should probably read that book, too. And not just because
mechu-deru was a mathematics-intensive power but probably not as much as IAC was. Here began the lengthy task of writing down the relevant sections to integral calculus. She knew the book would be a brick, and proper depth is important to ensure the textbook will work as intended.
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CHAPTER 8: INDEFINITE INTEGRALS AS THE REVERSE OF A DERIVATIVE
8.1. Differentials
Let us begin with the notion of the linear local approximation. In the neighborhood of x, the value of f(x+dx) can be approximated by f(x+dx) ~ f(x)+f'(x)dx: when dx is taken sufficiently small, the second term in f(x+dx) above is called the
differential of y.
Definition 8.1:
The
differential of x, noted dx, is a real number.
Definition 8.2:
Let y = f(x), where f(x) is a differentiable function. The
differential of y is dy = f'(x) dx.
[...] (a few examples of differentials and how to use it in practice)
8.2. Indefinite integral and basic formulae
In a first course in differential calculus, one calculated derivatives of functions. We shall now begin the learning of the reverse process: determine a function whose derivative is given.
Definition 8.3:
A function g is called an
antiderivative of a function f if
g'(x) = f(x)
Definition 8.4: An
indefinite integral of a function f(x) is an expression of the form g(x)+C where g(x) is an antiderivative of f(x) and C is a real number, noted
int f(x) dx = g(x)+C, if g'(x) = f(x)
C is called an
integration constant, f(x) the
integrand and the x of the dx expression denotes the integration variable.
Here is a list of a few elementary integration formulas:
int xn dx = xn+1/(n+1) + C (for n =/= -1)
int x-1 dx = ln x + C
int sin x dx = - cos x + C
int cos x dx = sin x + C
int sec2 x dx = tan x + C
int cosec2 x dx = - cot x + C
int sec x tan x dx = sec x + C
int cosec x cot x dx = - cosec x + C
int ax dx = ax / (ln a) + C
int (1-x2)-1/2 dx = arcsin x + C
int 1/(1+x2) dx = arctan x + C
int 1/[x(x2-1)] dx = arcsec x + C
Theorem 8.1:
If int fi(x) dx = gi(x)+Ci, then
sum_i=1^n ki int fi(x) dx = int sum_i=1^n kifi(x) dx
where the ki are real numbers.
[...] (Worked examples of indefinite integral calculations)
8.3. Integration by substitution
Sometimes it might be simpler to substitute a variable in another function's place, because the calculations may be lengthier to do otherwise. The substitution method consists to:
- Choose in the integrand a function f and to substitute u = f(x)
- Calculate the differential of u, du = f'(x) dx (also called the single-variable Jacobian)
- Express the initial integral as a function of this new variable u and the differential du
- Integrate as a function of this new variable u
- Express the answer as a function of the initial variable
(As we shall see in Chapter 15, in multivariate functions, the analog of the differential of u means that
du = |det Df(
x)|
dx where Df(
x) is the matrix of the partial derivatives of
u with respect to the various components of f(
x) and the determinant of the resulting matrix is called the Jacobian)
Theorem 8.2.: If G is a primitive of g, then
int g(f(x))f'(x) dx = G(f(x)) + C
[...] (The proof of Theorem 8.2 is given, and follows a few examples of calculating integrals using univariate Jacobians)
int tan x dx = ln |sec x| + C
int cot x dx = ln |sin x| + C
int sec x dx = ln |sec x + tan x| + C
int cosec x dx = ln |cosec x - cot x | + C
Theorem 8.3 (Integration by parts):
Let u and v be two differentiable functions of the same variable. Then int u dv = uv - int v du.
Proof: From the product rule,
d(uv) = u dv + v du
u dv = d (uv) - v du
int u dv = int d(uv) - int v du
int u dv = uv - int v du
[...] (a few examples of integration by parts)
8.4. Differential equations
Definition 8.5:
An
ordinary differential equation is an equation containing an unknown univariate function as well as one or several of its derivatives. Any function verifying a differential equation is called a
solution to the differential equation.
Some differential equations can be solved as follows:
- Separate the variables, that is, group everything dependent on one variable with its differential on each side of the equation; the differentials must be in the numerator
- Integrate each side of the equation
- Express one variable as a function of the other
Definition 8.6:
The condition f(x0) = y0 is called the
initial condition.
[...] (Several practical examples of solving differential equations)
CHAPTER 9: DEFINITE INTEGRALS AND THEIR APPLICATIONS
9.1. Summation
Definition 9.1:
The
summation is defined as follows:
sum_i=r^s ai = ar + ar+1 + ... + as-1 + as
where ai is the
summand, and the
index takes all the integer values between r, the
lower bound and s, the
upper bound, inclusive.
Theorem 9.1 (Properties of the summations):
sum_i=r^s (ai + bi) = sum_i=r^s ai + sum_i=r^s bi
sum_i=r^s cai = c sum_i=r^s ai (where c is a real number)
sum_i=r^s c = (s-r)c
[...] (a few examples of formulae making use of the summation symbol)
9.2: Definite integral
Definition 9.2:
A
partition P of [a,b] is a sequence of real numbers such as a = x0 < x1 < ... < xn-1 < xn = b. P is said to be
regular if xi+1 - xi is the same for all i.
Definition 9.3:
Let f be a continuous function over [a,b] and P a partition of [a,b]. A
Riemann sum is a sum of the form
sum_i=1^n f(ci) (xi-xi-1) where ci is part of [xi-1, xi].
In particular, if f(ci) is the minimum or the maximum of f(x) over [xi-1, xi], the resulting sum is known under the names of
lower sum and
upper sum respectively. In general, when f is continuous and nonnegative over [a,b], Riemann sums give an approximation of the actual area under the curve and the x-axis between a and b. Here f(ci) is regarded as the height of a rectangle of base length (xi-xi-1). To obtain a better approximation of the area, all that is needed is to decrease the base length which, in effect, increases the number of rectangles in the sum.
Definition 9.4:
Let f be a function defined over [a,b] and P a partition of [a,b]. The
definite integral of f over [a,b], noted
int_a^b f(x) dx =
lim sum_i=1^n f(ci) (xi-xi-1)
(max (xi-xi-1)) -> 0
where ci is part of [xi-1, xi], if the limit exists.
We shall see in Chapter 10 that, when the bounds are either infinity or where the integrand goes to infinity at a bound, we shall take the limit of the integral going to said bound and, in chapter 15, that the multivariate definite integral is defined as the limit of a similar but multiple summation where the limit of the meshes on
each of the integration variables go to 0.
Theorem 9.2:
If f is a continuous function over [a,b], then f is an
integrable function over [a,b].
[...] (The proof of theorem 9.2, as well as a myriad of applications of the definite integral)
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