Definite integrals

In a geometrical context the integral can be interpreted as the area between the graph of the integrand f(x) and the x-axis. The idea is to summarize infinitely many infinitely thin rectangles. (An alternative approach to areacomputation is the Lebesgue-integration where you partition the range of f(x) instead of the domain). This method works better for integration of infinite series e.g. Fourier series than the Riemann integral.

The integral sign is a stylized version of the letter ‘s’.

Bernhard Riemann discovered that this can be accomplished by forming one ‘uppersum’ and one ‘lowersum’.

An undersum can be seen here and to see the oversum click here.

Riemann showed that there exists exacctly one number which is smaller than the uppersum and bigger than the lower sum in the limit when the width of the rectangles approaches zero.

This number is identical to the integral over this interval.

Earlier in the 18th. century Newton and Leibniz discovered the fundamental theorem of calculus. This gives an easy method for computation of the integral with the aid of the primitive function

F'(x) =f(x) then $F(x) = \int_a^x f(t)\, dt\,.$

See the derivation of this theorem here in an excerpt from the Gamma textbook (björup et. al.) Fundamentaltheorem.

F'(x) = (F(x+dx) -F(x))/dx F’ (x) dx = (F(x+h) – F(x)).

By Summation of all these differential areaelements it can be seen that all terms cancel each other except the first and the last which gives us the fundamental theorem of calculus:

$F(b) - F(a) = \int_a^b f(x)\,dx,$