ALGEBRA INTEGRALS INTEGRATION
NOTES 2025
● An integral is a mathematical operation that represents the area under a curve or
the accumulation of a quantity over a given interval.
● An indefinite integral is a mathematical operation that represents the
accumulation of a quantity over an interval that does not have specific endpoints.
It is written in the form ∫f(x)dx, where f(x) is the function being integrated and dx
represents the infinitesimally small interval over which the accumulation is taking
place.
● A definite integral is a mathematical operation that represents the accumulation
of a quantity over a specific interval between two points. It is written in the form
∫a^bf(x)dx, where a and b are the specific endpoints of the interval, f(x) is the
function being integrated, and dx represents the infinitesimally small interval over
which the accumulation is taking place.
● To find an indefinite integral, the function being integrated is rewritten in a specific
form and an antiderivative is found. An antiderivative is a function that, when
differentiated, gives the original function. For example, if the indefinite integral of
f(x) is represented as ∫f(x)dx, then the antiderivative of f(x) is represented as F(x),
where F'(x) = f(x).
● To find a definite integral, the area under the curve between two specific points is
calculated using the indefinite integral and the specific end points. For example,
to find the definite integral of f(x) between the points a and b, the indefinite
integral of f(x) is evaluated at the point a, and then evaluated again at the point b.
The difference between these two values is the definite integral of f(x) between
the points a and b.
● The Fundamental Theorem of Calculus states that the definite integral of a
function over a given interval is equal to the difference between the
antiderivatives of the function at the endpoints of the interval. This theorem is a
fundamental principle of calculus that connects the concept of integration with the
concept of differentiation.
● Integration can be used to solve a variety of problems, including finding the area
under a curve, calculating the volume of a solid, and solving differential
equations. Some common applications of integration include calculating the
length of a curve, finding the volume of a cylinder or sphere, and determining the
