Lecture III Indefinite integral. Definite integral

Lecture questions Antiderivative Indefinite (primitive) integral Indefinite integral properties Formulas of integrating some functions Curvilinear trapezoid. Area of a curvilinear trapezoid. Riemann Sum Definite integral Fundamental Theorem of Integral Calculus Newton – Leibniz formula

Antiderivative. Indefinite integral

Antiderivative Antidifferentiation (integration) is the inverse operation of the differentiation. In calculus, an antiderivative of a function f(x) is a function F(x) whose derivative is equal to f(x) F ′(x) = f(x) or dF=f(x)dx

Antiderivative Any constant may be added to F(x) to get the antiderivative of the function f(x). Antidifferentiation (or integration) is the process of finding the set of all antiderivatives of a given function f(x)

Antiderivative The entire antiderivative family of f(x) can be obtained by changing the value of C in F(x); where C is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's location depending upon the value of C.

Indefinite integral Terminology: - integral symbol x – integration variable f(x) - integrand (subintegral function) f(x)dx - integrand (integration element) C – constant of integration

Integral properties The first derivative of the indefinite integral is equal to subintegral function: The differential of the indefinite integral is equal to integration element: The general antiderivative of a constant times a function is the constant multiplied by the general antiderivative of the function (The constant multiple rule): If f(x) and g(x) are defined on the same interval, then:

Formulas of integrating of some functions

Techniques of integration Method of direct integration using integral formulas and properties Integration by substitution Integration by Parts

Definite integral

Curvilinear trapezoid The figure, bounded by the graph of a function y=f(x), the x-axis and straight lines x=a and x=b, is called a curvilinear trapezoid.

Area of a curvilinear trapezoid. Riemann Sum

Definite integral The smaller the lengths Δx i of the subintervals, the more exact is the above expression for the area of the curvilinear trapezoid. In order to find the exact value of the area S, it is necessary to find the limit of the sums S n as the number of intervals of subdivisions increases without bound and the largest of the lengths Δx i tends to zero.

Fundamental Theorem of Integral Calculus. Newton – Leibniz formula If f(x) is continuous and F(x) is any arbitrary primitive for f(x) i.e. any function such that then

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