Integration?

Question

Does anyone know how to derive the formula that states the integral = area under the graph?

Answer 1

The area under the graph (curve) is almost the definition of the definite integral (as opposed to the indefinite integral). The Riemann definition of a definite integral is the infinite limit of the sum of the areas of a set of adjacent rectangles as the number of them goes to infinity while their widths go to zero. For any finite number (and non-infinitesimal widths) it's easy to see that the sum of their areas is an approximation of the area of the smooth/continuous curve and that the approximation gets better the larger the number of rectangles until the two areas are equal in the infinite limit.

This is also closely related to the "Fundamental Theorem of Calculus", which links the integral to the derivative - the two fundamental operations in calculus.

See the Wikipedia web pages that I put links to under "Sources" below. They have nice derivations/proofs and explanations with pictures.

Answer 2

If the function is a step function, it's obvious. The integral is just the sum of the areas of the rectangles between the graph and the axis. In the general case, you just approximate a general integrable function by step functions and take the limit.

Answer 3

Is it between 2 points? If so, find the antiderivative of the equation of the graph. Then find the difference between the antiderivative at point B and point A.