A Riemann sum approximates the area under a curve using rectangles of varying widths, where the width of each rectangle is the interval of the partition and the height is the value of the function at that point. It allows calculating the area under a curve or length of a graph by breaking it into smaller pieces. The example shows using a left Riemann sum with 4 subintervals to estimate the area under the parabola f(x) = x^2 + 1 from 0 to 4.

1. A Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after
nineteenth century German mathematician Bernhard Riemann. One very common application is
approximating the area of functions or lines on a graph, but also the length of curves and other
approximations.
EXAMPLE 1: Find the area of the region bounded by 𝑓(𝑥) = 𝑥2
+ 1 on an interval [0,4] using L4
SOLUTION: Note: In getting the area of the rectangle the formula you used 𝑎𝑟𝑒𝑎 = 𝐿 𝑥 𝑊.
Length With Area
1st
1 1 1
2nd
2 1 2
3rd
5 1 5
4th
10 1 10
Final Answer 𝟏𝟖 𝒖𝒏𝒊𝒕𝟐
Under Estimated

2. Finding the length:
𝑓(0) = (0)2
+ 1 = 1 (1st
rectangle)
𝑓(1) = (1)2
+ 1 = 2 (2nd
rectangle)
𝑓(2) = (2)2
+ 1 = 5 (3rd
rectangle)
𝑓(3) = (3)2
+ 1 = 10 (4th
rectangle)