Riemann Sum

Basic Idea of a Riemann Sum

Imagine a plot of land with one irregular curved edge. To estimate its area, we can divide the plot into rectangular strips of equal width. We then calculate the area of each strip and add the results.

A Riemann sum divides an interval into subintervals, chooses one sample point in each, and adds the signed rectangle contributions. For a Riemann-integrable function, these sums approach the definite integral as the widest subinterval approaches zero. When the function is nonnegative, the signed accumulation equals the area under the curve. With equal-width partitions, increasing the number of rectangles makes each one narrower, although one additional rectangle does not always move the sum closer to the definite integral.

Symbol	Meaning
$[a,b]$	Interval divided into subintervals
$n$	Number of subintervals
$\Delta x=\frac{b-a}{n}$	Width of each subinterval
$x_i^*$	Sample point in subinterval $i$ , which gives the rectangle height $f(x_i^*)$

Riemann Sum

Basic Idea of a Riemann Sum

Riemann Sum

Basic Idea of a Riemann Sum

Interval Partition and Width

The Riemann Sum Formula

Left Endpoint Sum for a Linear Function

Exercises

Worked Solution