Riemann Sums
Introduction to Riemann Sums
Riemann sums are a method for approximating the area under a curve, which is equivalent to the definite integral of a function. They work by dividing the area into multiple rectangles and summing their areas.
For a function $f(x)$ on an interval $[a, b]$, we divide the interval into $n$ subintervals of equal width:
$$\Delta x = \frac{b - a}{n}$$
The Riemann sum is then defined as:
$$\sum_{i=1}^{n} f(x_i) \cdot \Delta x$$
Where $x_i$ is a point in the $i$-th subinterval.
Different types of Riemann sums are defined based on how we choose $x_i$:
- Left Riemann Sum: $x_i$ is the left endpoint of each subinterval
- Right Riemann Sum: $x_i$ is the right endpoint of each subinterval
- Midpoint Riemann Sum: $x_i$ is the midpoint of each subinterval
As the number of subintervals approaches infinity, all types of Riemann sums converge to the definite integral:
$$\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \cdot \Delta x = \int_{a}^{b} f(x) \, dx$$
Interactive Riemann Sum Visualizer
Approximation Result:
The approximated area using midpoint Riemann sum with 10 rectangles: 0
The exact area (definite integral): 0
Error: 0 (0%)
Understanding Convergence
As you increase the number of rectangles in the Riemann sum, you'll notice that the approximation gets closer to the exact value of the definite integral. This demonstrates the key idea behind integral calculus: we can compute the exact area under a curve by taking the limit of these approximations as the number of rectangles approaches infinity.
For a continuous function $f(x)$ on a closed interval $[a,b]$, we have:
$$\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$$
where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is any point in the $i$-th subinterval.
This concept forms the basis of the Fundamental Theorem of Calculus, which we'll explore in another section.
Notes and Resources
Key Takeaways
- Riemann sums provide a way to approximate the area under a curve by dividing it into rectangles
- Different types of Riemann sums (left, right, midpoint) offer different approximation strategies
- As the number of rectangles increases, the approximation becomes more accurate
- The limit of Riemann sums as the number of rectangles approaches infinity gives the definite integral
- Riemann sums form the foundation for understanding integration in calculus