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:   

About the activity

Remarkably, integration is precisely the opposite of differentiation—a result known as the Fundamental Theorem of Calculus.

To see why this should be so, remember that the definite integral is an area function. So, given a curve with equation \(y=f(x)\) and associated area function \(A(x)\), let's consider a small step further along the \(x\)-axis and the corresponding increase in area, described mathematically by $$A(x + \Delta x) = A(x) + \Delta A,$$ and as shown in the following figure.

But what is \(\Delta A\)? Well, if \(\Delta x\) is small, then the function \(f(x)\) will not have changed too much, and just like we saw in the above activity, we can estimate this additional area using a little box of height \(f(x)\) and base \(\Delta x\) (shaded yellow in the diagram), giving $$A(x + \Delta x) \approx A(x) + f(x) \Delta x.$$

Similarly to what we saw for differentiation, in the limit \(\Delta x \rightarrow 0\) this becomes an equality and so $$\lim_{\Delta x \rightarrow 0} \frac{A(x + \Delta x) - A(x)}{\Delta x} = f(x).$$ The expression on the left hand side is just the derivative of the area function \(A(x)\), and we know that the area function is the definite integral, hence the derivative of the integral is just the original function!

Mathematically this corresponds to the beautiful and quite fancy equation: $$\frac{d}{dx}\left(\int_a^xf(t)\,dt\right) = f(x),$$ which simply says:

If you differentiate the integral of a function, you end up right back where you started.
They are inverse operations.

Riemann is also famous for important work he did with prime numbers, in particular relating to the Prime Counting Function, which says the number of prime numbers less than \(x\) is approximately given by the formula \(\pi(x) \approx \frac{x}{\log x}\). This implies that if you choose some random positive integer \(n\), the probability that it is prime is about \(\frac{1}{\log n}\). Can you see why? Riemann's related function is known as the Zeta Function, and finding the roots of this function is one of the most important unsolved problems in mathematics today.