About the Riemann Sums Calculator

In calculus, finding the area under a curve is a central problem known as integration. Before we learn the fundamental theorem of calculus, we approximate this area using Riemann Sums. This technique involves slicing the area into narrow rectangles and summing their areas.

How it Works

To approximate the area under f(x) from a to b:

1. Divide the interval [a, b] into n subintervals of equal width Δx = (b-a)/n.
2. Choose a sample point x* in each subinterval.
3. Form a rectangle with width Δx and height f(x*).
4. Sum the areas: Area ≈ Σ f(x*)Δx.

The Three Common Methods

• Left Riemann Sum: The height is determined by the value of the function at the left endpoint of each subinterval. It underestimates increasing functions.
• Right Riemann Sum: The height is determined by the value at the right endpoint. It overestimates increasing functions.
• Midpoint Rule: The height is determined by the value at the center of the subinterval. This is usually the most accurate of the basic rectangle methods.

Why use a Riemann Sums Calculator?

Manually calculating the area of 10 or 20 rectangles is repetitive and error-prone arithmetic. This tool automates the summation process, allowing you to experiment with increasing n to see how the approximation approaches the true integral. Visualizing the rectangles helps build the intuition for the definition of the definite integral.