This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed by i, each of which is "tagged" with a distinguished point ti ε [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as
Riemann integral
Approximations to integral of √x from 0 to 1, with ■ 5 right samples (above) and ■ 12 left samples (below)
Riemann sums
Riemann sums converging as intervals halve, whether sampled at ■ right, ■ minimum, ■ maximum, or ■ left.
Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral
is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.
Measuring the area under a curve
Definite Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
Definite integral of a function
The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation as:
Fundamental Theorem of Calculus
Let f(x) be a continuous function in the given interval [a, b], and F is any anti-derivative of f on [a, b], then
Area between two curves y = f(x) and y = g(x)
DEFINITION
If f and g are continuous and f (x) ≥ g(x) for a ≤ x ≤ b, then the area of the region R between f(x) and g(x) from a to b is defined as
Area between two curves y = f(x) and y = g(x) Examples
Definite Integrals to find the Volumes
We can also use definite integrals to find the volumes of regions obtained by rotating an area about the x or y axis.
