Integrals and primitives with a senior math course free download PDF.
In this lesson we will see the definition and the different properties of the integral as well as the geometrical meaning with the areas. In addition, in this course you will see the different ways to calculate an integral using the primitive and the associativity and linearity properties of the integral. In addition,integrals and primitives must be mastered in order to progress well in math .
I. Definitions and properties of the integral and primitives.
In particular, functions which are derivable on an interval I, are integrable on I, but this condition, although sufficient, is not necessary. The function f defined on by , although not derivable at 0, is integrable on , because it has as primitive on the function F such that: .
However, except in special cases containing indications, in the baccalaureate problems, functions integrable on an interval I will always be functions derivable on I.
then whatever a I and b I, we have
F(b) – F(a) = G(b) – G(a)Indeed, if F ‘(x) = G ‘(x), then there exists c such that G(x) = F(x) + c.
Therefore: G(b) – G(a) = F(b) + c – [F(a) + c] = F(b) – F(a).
The difference in the images of b and a for any primitive of f is the same.
This number depends only on f, a and b.
This will allow us to give the following definition:
Let a I and b I be two real numbers of this interval I,
The integral from a to b of the function f is the number F(b) – F(a) where F is any primitive of f on I.
which reads“sum from a to b of f from t dt ” and is also called“integral of f between a and b“.
In the writing , the letter t is called: “dumb variable”.
Indeed, we can also write The letter “dummy variable” indicates the name of the “variable of integration” (any other letter in the expression of the function f to be integrated is then considered as constant. The interest of this appears when there are several variables, but this is not in the Terminale program, however, it will be useful in the presence of parameters).
is also written in the condensed form using F :
The function is derivable on .
It is therefore integrable on and admits primitives on .
For example is a primitive of f on .
II. consequences of the definitionThe first properties.
Let f be a function derivable on an interval [a,b].
From this property, we deduce an important synthesis that links a derivable function, its derivative function and the notion of integral.
Then, for all x I, we have .
III.primitive function of an integrable function.
the function defined on I by is the primitive of f which cancels at x = a.
If G is any primitive of f on I, then , so
Fa‘(x) = G ‘(x) = f(x) .
Indeed, G(a) is a constant, so its derivative is zero and G ‘(x) = f(x) because G is a primitive of f. Conclusion: is also a primitive of f .
Moreover:Fa (a) = G(a) – G(a )= 0, soFa cancels for x=a .
Compute the following integrals, then state the primitives they define.
IV. Integral of a positive function.
Call F a primitive of f on [a;b]Then we have F ‘(x) = f(x) for all x [a;b]The derivative f of F being positive on [a;b]the function F is increasing on [a;b].as a < b, then we have: F(a) F(b).
V. Integrals and areas
1. summary table
a x b and 0 y f(x)Then Area of D = units of areaThe unit of area being that of the rectangle whose sides are the units of length of the abscissa and ordinate.
The set D consists of the points located between the curve representing the function f, the x-axis and the lines of equation x = a and x = b.
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