A Riemann sum, named after the 19th century German mathematician Bernhard Riemann, is a method for calculating the area under a smooth curve by approximating it with a series of equal width rectangular boxes. The area of the box at coordinate $$x=x_i$$ is the height, $$f(x_i)$$ times the base, $$\Delta x$$, and so the $$n$$-box approximation to the total area is given by the sum: $$A(x)\approx\sum_{i=1}^n f(x_i)\Delta x.$$

As the width of the boxes becomes narrower, the sum of their areas gets closer and closer to the area under the curve. Just like we saw in Introducing the Derivative, in the limit that the rectangles become infinitely thin (and thus of zero area themselves), we get the somewhat paradoxical result that their area sums to exactly to the area under the curve.

Integration is the mathematical process of accumulating these infinitesimally small areas, and the name for the (signed) area under the curve between two points $$x=a$$ and $$x=b$$ is the definite integral of the function $$f(x)$$, written as $$A(x)=\int_a^bf(x)\,dx.$$

The activity below demonstrates how in the limit $$\Delta x \rightarrow 0$$ the approximation to the area improves as the box width decreases.

Introducing the Integral
“A box without hinges, key, or lid, yet golden treasure inside is hid.”
Bilbo Baggins

Enter a function and select an approximation rule—Riemann sums or the Trapezoidal rule. Change the number of subdivisions and see how the estimate improves.

Entry of functions is the same as for the Function Viewer activity. Scale with the sliders, and drag the background to pan.

Function:

Rule:     Subdivisions:

x scale:       y scale: