What is a Riemann Sum?
In mathematics, a Riemann Sum is a method for approximating the total area underneath a curve on a graph. This concept serves as the foundational building block for the definite integral in calculus.
Because calculating the exact area of complex curves is difficult, Riemann sums break the area down into easily measurable geometric shapes—usually rectangles or trapezoids. By calculating the area of these individual shapes and adding them together, we get a reliable estimation of the total area.
Left, Right, and Midpoint Rules
When using rectangles to estimate the area, you must decide how tall each rectangle should be. The height is determined by evaluating the function $f(x)$ at a specific point within each subinterval:
- Left Riemann Sum: Uses the left endpoint of the subinterval to determine the height. It may underestimate or overestimate depending on whether the function is increasing or decreasing.
- Right Riemann Sum: Uses the right endpoint. Like the left sum, its accuracy depends on the curve's slope.
- Midpoint Rule: Uses the exact middle point of the subinterval. This is generally more accurate than Left or Right sums because overestimation and underestimation cancel each other out slightly.
The Formula and Delta X ($\Delta x$)
The first step in any Riemann sum calculation is determining the width of your rectangles, known as $\Delta x$. The formula is:
Where b is the upper bound, a is the lower bound, and n is the number of subintervals. As n approaches infinity, $\Delta x$ approaches zero, and the Riemann Sum becomes the exact Definite Integral.