Module 10 the reimann sums

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)