Integration Advanced

Mean Value Theorem for Integrals

Area Between Two Curves

Volume of Solid Shapes

Arc Length

What is the Riemann Sum ?

The Riemann Sum only gives an approximation
We must take the limit of all the Riemann sums to find the exact value

The Riemann Sum is a method for approximating the total area underneath a curve on a graph.
Some of the technical deficiencies in the Riemann Integral can be remedied by the Riemann Stieltjes Integral and most disappear with the Lebesgue Integral.

What is the Riemann Integral ?

The Riemann Integral can be thought of as the limit of the Riemann Sums of a function as the partitions get smaller.
If the limit exists then the function is said to be Riemann integrable.
The Riemann Sum can be made as close as you want to the Riemann Integral by making the partition small enough.

What is the Riemann Stieltjes Integral ?

This is a generalisation of the Riemann Integral
The R-S integral of a real-valued function f of a real variable with respect to a real function g is denoted by:

What is the Lebesgue Integral ?

The general theory of integration of a function with respect to a general measure
The specific case of integration of a function defined on a sub domain of the real line or a higher dimensional Euclidean Space with respect to the Lebesgue measure

InDefinite Integral

The indefinite integral is the antiderivative, the inverse operation to the derivative

Definite Integral

The definite integral inputs a function and outputs a number which gives the area between the graph of the input and the x axis
The technical definition of the definite integral is the limit of a sum of areas of rectangles called the Riemann Sum

L'hopitals Rule

This is a great shortcut for doing limit problems
Taking the limit with L'hopitals Rule
Don't use the Quotient Rule; just take the derivatives of the of the numerator and denominator separately
The rules lets you replace the numerator and denominator with their corresponding derivatives

Let f and g be differentiable functions
If the limit of f(x)/g(x) as x approaches c produces 0/0 or infinity/infinity then
lim f(x)/g(x) = lim f'(x)/g'(x)