Calculus 2, part 1 of 2: Integrals with applications
Integral calculus in one variable: theory and applications for computing area between curves, curve length, and volumes
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57 hours
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Oct 2024
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What you will learn
How to solve problems concerning integrals of real-valued functions of 1 variable (illustrated with 419 solved problems) and why these methods work.
The concept of antiderivative / primitive function / indefinite integral of a function, and computing such integrals in a process reverse to differentiation.
Integration by parts as the Product Rule in reverse with many examples of its applications.
Integration by substitution as the Chain Rule in reverse with many examples of its applications.
Integration of rational functions with help of partial fraction decomposition.
Various types of trigonometric integrals and how to handle them.
Direct and inverse substitutions; various types of trigonometric substitutions.
The tangent half-angle substitution (universal trigonometric substitution).
Euler's substitutions.
Triangle substitutions.
Riemann integral (definite integral): its definition and geometrical interpretation in terms of area.
An example of a function that is not Riemann integrable (the characteristic function of the set Q, restricted to [0,1]).
Oscillatory sums; Cauchy criterion of (Riemann) integrability.
Sequential characterisation of (Riemann) integrability.
Proof of uniform continuity of continuous functions on a closed bounded interval.
Integrability of continuous functions on closed intervals.
Integration by inspection: Riemann integrals of odd (or: even) functions over compact and symmetric-to-zero intervals.
Integration by inspection: evaluating some definite integrals with help of areas known from geometry.
Fundamental Theorem of Calculus (FTC) in two parts, with a proof.
Applications of Fundamental Theorem of Calculus in Calc 2 and Calc3.
Application of FTC for computing derivatives of functions defined with help of Riemann integrals with variable (one or both) limits of integration.
Application of FTC for computing limits of sequences that can be interpreted as Riemann sums for some integrable functions.
The Mean-Value Theorem for integrals with proof and with a geometrical interpretation; the concept of a mean value of a function on an interval.
Applications of Riemann integrals: (signed) area between graphs of functions and the x-axis, area between curves defined by two continuous functions.
Applications of Riemann integrals: rotational volume.
Applications of Riemann integrals: rotational area.
Applications of Riemann integrals: curve length.
Improper integrals of the first kind (integration over an unbounded interval).
Improper integrals of the second kind (integration of unbounded functions).
Comparison criteria for determining whether an improper integral is convergent or not.
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udemy ID
3/3/2024
course created date
10/9/2024
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