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The
definite integral |
The area between the graph
of a function and the x-axis over a closed interval
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The upper sum and the lower sum |
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The definition of the definite integral |
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Riemann
sum |
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Calculating
a definite integral from the definition |
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Calculating
a definite integral from the definition, examples |
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Physical
applications to the definite integral |
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Describing
motion of objects using velocity - time graphs |
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Evaluating
the area under the graph of a function using the definition of the
definite integral, examples |
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The
definite and indefinite integrals |
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The area between the graph
of a function and the x-axis over a closed interval |
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Geometric interpretation of the definite integral |
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Properties of the definite integral |
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The definite integral
over interval of zero length |
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Reverse
order of integration |
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The constant multiple rule |
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The integral of the sum or difference of two functions |
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Internal addition of the definite integral |
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The definite integral of an odd function |
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The definite integral of an even function |
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The definite integral of a nonnegative and nonpositive functions |
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Relationship of the definite integrals of two functions over the same
interval of integration |
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Absolute integrability |
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The mean value theorem |
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The average value of a function over the given interval |
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The indefinite integral |
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Integration - inverse of differentiation |
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Constant
of integration, the arbitrary constant term in the expression of the
indefinite integral |
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Evaluating the indefinite integral example |
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The fundamental theorem of
calculus |
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The fundamental theorem of differential calculus
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Evaluating definite integrals
using indefinite integrals |
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The fundamental theorem of integral calculus |
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Evaluating the definite integral examples |
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a)
The
indefinite integral |
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Basic
rules of integration |
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Table
of indefinite integrals |
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The indefinite integral applications |
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Kinetic equation of free fall |
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Exponential
growth and decay (increase and decrease) |
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Evaluation of indefinite integrals using some basic integration rules
and formulas, examples |
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b)
The
indefinite integral |
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Substitution rule |
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Evaluating the indefinite integrals using substitution rule, examples
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Evaluating the indefinite integrals using substitution rule,
solutions
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Integration by parts rule |
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Evaluating the indefinite integrals using the integration by parts
formula, examples
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Evaluating the indefinite integrals using the integration by parts
formula, solutions
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The indefinite integrals
containing quadratic polynomial (trinomial)
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The indefinite integrals
containing quadratic polynomial, examples
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Recursion
formulas - use
of integration by parts
formula
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Recursion
formulas - use
of
integration by parts
formula, examples
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c)
The
indefinite integral |
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Integrating
rational functions
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Use
of the
partial fraction decomposition to integrate a proper rational
function
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Use
of the
partial fraction decomposition to integrate a proper rational
function, examples
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The
Ostrogradsky method of the integration of a proper rational
functions
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The
Ostrogradsky method of the integration of a proper rational
functions, examples
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Integrating
irrational functions
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Integration of
irrational functions, examples
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Integration
of irrational functions of the form
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Integration
of irrational functions of the form
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Euler's
substitutions
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Integrating
irrational functions using Euler's substitutions, examples
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Binomial
integral
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Solving
binomial integrals, examples
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d)
The
indefinite integral |
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Trigonometric
integrals
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Trigonometric
integrals of the form
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∫
sinm x, cosn x dx
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Trigonometric
integrals of the form
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∫
sin (m x) sin (n x) dx,
∫
sin
(m x)
cos (n x)
dx,
∫
cos (m x)
cos (n x)
dx.
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Integrals
of the rational functions containing sine and cosine,
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∫
R
(sin x, cos x) dx
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Integrals
of the hyperbolic functions
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Substitution
and definite integration
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Substitution
and definite integration examples
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Integrations by parts and the definite integral
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The improper integrals
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The improper integral definition
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The improper integrals examples
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Differentiation
and integration of infinite series
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Differentiation
and integration of infinite series examples
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Applications
of the definite integral
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The
area of a region in the plane
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The
area between the graph of a curve and the coordinate axis
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The
area between the graph of a curve and the coordinate axis examples
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The
area bounded by a parametric curve
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The
area of the sector of a curve in polar coordinates
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The
area of the sector of a curve given in Cartesian (or rectangular) coordinates
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The
area of the sector of a parametric curve
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The
area between two curves
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The
area between two curves examples
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Length
of plane curve, arc length
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Arc length
of a parametric curve
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Arc length of a curve in polar coordinates
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The
volume
of a solid of revolution
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The
volume
of a sphere
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The
volume
of a spherical segment
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The
volume
of a cone
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The
surface area of a solid of revolution
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The
lateral surface area of a cone
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The surface area of a spherical
cap
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The surface area of an
ellipsoid
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