Riemann sums 2
Riemann sums 2
April 13, 2018
On your cell phone, go to your play store. Download and install SMART lab Open Enter your name and the class ID: 0543 9926
Riemann Sums
1
Riemann sums 2
April 13, 2018
In mathematics, 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.
Suppose we want to find the exact area under this curve!
2
Riemann sums 2
April 13, 2018
You can use rectangles to find the area. The more rectangles that you use, the closer you get to the actual answer.
These sorts of approximations are called Riemann sums, and they're a foundational tool for integral calculus.
When all of the rectangles touch the curve with the left top corner, this is called a Left Riemann Sum.
3
Riemann sums 2
April 13, 2018
When all of the rectangles touch the curve with the right top corner, this is called a Right Riemann Sum.
4
Riemann sums 2
April 13, 2018
Terms commonly mentioned when working with Riemann sums are "subdivisions" or "partitions." These refer to the number of parts we divided the xinterval into, in order to have the rectangles.
Simply put, the number of subdivisions (or partitions) is the number of rectangles we use.
5
Riemann sums 2
April 13, 2018
Subdivisions can be uniform, which means they are of equal length, or nonuniform.
6
Riemann sums 2
April 13, 2018
Imagine we're asked to approximate the area between y=g(x) and the xaxis from x=2 to x=6 using a Left Riemann Sum.
Find the area of each rectangle
7
Riemann sums 2
April 13, 2018
Area is approximately 20u2
8
Riemann sums 2
April 13, 2018
Buuuuuuuuuuuuuuuut...........
What if we don't have a graph???????
Imagine we're asked to approximate the area between the xaxis and the graph of f from x = 1 to x = 10 using a right Riemann sum with three equal subdivisions. Use the table below to help find your answers.
9
Riemann sums 2
April 13, 2018
1. Divide the interval into n different rectangles. 2. Evaluate each end point at that x value. 3. Determine if you are using left Riemann or right Riemann or if you are trying to find the interval in which the answer will lie. 4. Find the area of each rectangle and add the sums together.
x = 1 to x = 10
Length = 3
Height = ___ Height = ___ Height = ___
10
Riemann sums 2
April 13, 2018
Approximate the area between the xaxis and the graph of f(x) = 2x from x =3 to x = 3 using a right Riemann sum with three equal subdivisions.
11
Riemann sums 2
April 13, 2018
12
Riemann sums 2
April 13, 2018
13
April 13, 2018
On your cell phone, go to your play store. Download and install SMART lab Open Enter your name and the class ID: 0543 9926
Riemann Sums
1
Riemann sums 2
April 13, 2018
In mathematics, 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.
Suppose we want to find the exact area under this curve!
2
Riemann sums 2
April 13, 2018
You can use rectangles to find the area. The more rectangles that you use, the closer you get to the actual answer.
These sorts of approximations are called Riemann sums, and they're a foundational tool for integral calculus.
When all of the rectangles touch the curve with the left top corner, this is called a Left Riemann Sum.
3
Riemann sums 2
April 13, 2018
When all of the rectangles touch the curve with the right top corner, this is called a Right Riemann Sum.
4
Riemann sums 2
April 13, 2018
Terms commonly mentioned when working with Riemann sums are "subdivisions" or "partitions." These refer to the number of parts we divided the xinterval into, in order to have the rectangles.
Simply put, the number of subdivisions (or partitions) is the number of rectangles we use.
5
Riemann sums 2
April 13, 2018
Subdivisions can be uniform, which means they are of equal length, or nonuniform.
6
Riemann sums 2
April 13, 2018
Imagine we're asked to approximate the area between y=g(x) and the xaxis from x=2 to x=6 using a Left Riemann Sum.
Find the area of each rectangle
7
Riemann sums 2
April 13, 2018
Area is approximately 20u2
8
Riemann sums 2
April 13, 2018
Buuuuuuuuuuuuuuuut...........
What if we don't have a graph???????
Imagine we're asked to approximate the area between the xaxis and the graph of f from x = 1 to x = 10 using a right Riemann sum with three equal subdivisions. Use the table below to help find your answers.
9
Riemann sums 2
April 13, 2018
1. Divide the interval into n different rectangles. 2. Evaluate each end point at that x value. 3. Determine if you are using left Riemann or right Riemann or if you are trying to find the interval in which the answer will lie. 4. Find the area of each rectangle and add the sums together.
x = 1 to x = 10
Length = 3
Height = ___ Height = ___ Height = ___
10
Riemann sums 2
April 13, 2018
Approximate the area between the xaxis and the graph of f(x) = 2x from x =3 to x = 3 using a right Riemann sum with three equal subdivisions.
11
Riemann sums 2
April 13, 2018
12
Riemann sums 2
April 13, 2018
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