Integrals are Fun: Illustrated Riemann-Stieltjes Integral. Illustrated examples of the Riemann-Stieltjes Integration in Python with Matplotlib.

*Integrals* are more than just the sum of its parts! Well, let’s not exaggerate. In their most fundamental definition, they are *only a sum* of infinitely many rectangles under the curve of some function **f** over some interval **[a, b]** for **a, b ∈ ℝ**. However, solving and numerically approximating them is lots of fun and besides that, they are quite useful in applied sciences.

In the previous blog dedicated to the integrals, we went through the most basic form of integration, namely the **Riemann Integral**. This time we take a step forward and learn about its more general extension, the Riemann-Stieltjes Integral, the precursor of the great **Lebesgue Integral**.

In the following few minutes of reading, first, we give intuitive reasoning why the Riemann Integral is limited and how to escape those restrictions. This paves the way to the formal definition of the **Riemann-Stieltjes Integration**. Taking the theoretical foundations for granted we provide a straightforward *Python* implementation. Finally, we make an illustration using Matplotlib to facilitate the understanding of this subject. **Stay tuned!**

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