lecture iii indefinite integral. definite integral
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Lecture IIIIndefinite integral. Definite integral
Lecture questionsAntiderivativeIndefinite (primitive) integral Indefinite integral propertiesFormulas of integrating some functionsCurvilinear trapezoid. Area of a curvilinear trapezoid. Riemann SumDefinite integralFundamental 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)
AntiderivativeAny 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)
AntiderivativeThe 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 integralTerminology: - integral symbolx integration variablef(x) - integrand (subintegral function)f(x)dx - integrand (integration element)C constant of integration
Integral propertiesThe 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 integrationMethod of direct integration using integral formulas and propertiesIntegration by substitutionIntegration by Parts
Curvilinear trapezoidThe 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 integralThe smaller the lengths xi 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 Sn as the number of intervals of subdivisions increases without bound and the largest of the lengths xi tends to zero.
Fundamental Theorem of Integral Calculus.Newton Leibniz formulaIf f(x) is continuous and F(x) is any arbitrary primitive for f(x) i.e. any function such that
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