Blog Archives


Illustration of volume under a surface (double...

Image via Wikipedia

Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f is written:

\int_E f(x) \, dx.

Here x need not be a real number, but can be another suitable quantity, for instance, a vector in R3. Fubini’s theorem shows that such integrals can be rewritten as an iterated integral. In other words, the integral can be calculated by integrating one coordinate at a time.

Read the rest of this entry


Plot of 1/(sqrt(x)*(x+1)) from 0.093 to 3.0

Image via Wikipedia

A “proper” Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals.

If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity.

Read the rest of this entry


Plot of approximations to integral of sqrt(x) ...

Image via Wikipedia

A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell).

  • Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that mf (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(ba) and M(ba), it follows that
 m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).
Read the rest of this entry


Square root of x formula. Symbol of mathematics.

Image via Wikipedia

The collection of Riemann integrable functions on a closed interval [a, b] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration

 f \mapsto \int_a^b f \; dx

is a linear functional on this vector space. Thus, firstly, the collection of integrable functions is closed under taking linear combinations; and, secondly, the integral of a linear combination is the linear combination of the integrals,

Read the rest of this entry

%d bloggers like this: