The concept of the definite integral in calculus can feel abstract at first. Riemann sums provide a step-by-step way to make sense of it. By approximating the area under a curve using rectangles, we can see how increasing the number of rectangles makes the approximation more accurate. Eventually, this leads to the exact area, which is the value of the integral.
On this page, you can interactively explore Left Riemann Sums, Right Riemann Sums, and the Midpoint Rule. Adjust the number of subintervals and watch how the rectangles change. This will help you develop a clear understanding of how integrals emerge as the limit of Riemann sums.
Whether you are a student learning calculus for the first time, a teacher looking for an educational tool, or just curious about mathematics, this visualization will give you an intuitive grasp of how integrals work.
The integral of a function can be defined as the limit of Riemann sums as the number of subintervals approaches infinity:
\[ \int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x \]
where \(\Delta x = \dfrac{b - a}{n}\) and \(x_i^*\) is a sample point in the \(i\)th subinterval.
This visualization demonstrates how different types of Riemann sums approximate the area under a curve:
Instructions: Select a function and Riemann sum type, then adjust the interval [a, b] and the number of subintervals to see how the Riemann sum approximates the integral. Observe how increasing the number of subintervals improves the approximation.
Try: Select the sine function with interval [-p, p] to see both positive and negative rectangles.