Lesson Plan #6
Lesson Plan #50
Class: AP Calculus Date: Thursday January 12th, 2012
Topic: Riemann Sums Aim: How do we use Riemann Sums to approximate the area under a curve?
Objectives:
1) Students will be able to evaluate Riemann Sums.
HW# 50: Show the work involved in solving this example:
Using right endpoints, approximate the area between the curve [pic]and the x-axis bounded by the [pic]and [pic]using 5 rectangles.
Do Now:
1) Find the area of the region bound by[pic], [pic],[pic]and the [pic].
2) The graph of a piece-wise linear function[pic], for [pic], is shown above. What is the value of [pic]
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Assignment #1:
In the figure at right, do we have a figure, whose area we can find, that we can fit exactly under the curve?
We can’t find the exact area by fitting a figure, whose area we can find, that can fit exactly under the curve. So let’s try approximating the area. How? Let’s fit as best as we can a shape whose area we do know how to calculate. Let’s try rectangles!
How could we find the area of the 6 rectangles?
Let’s see!
To calculate the area of each rectangle we need the width and the height of each rectangle.
In this case the width is 1. In general, you take the higher number minus the lower number and divide by the number or rectangles to give you the width of each rectangle. In this case it is
[pic]. The height of each rectangle is obtained by taking [pic]value of the left endpoint of each rectangle and substituting into the function.
So we get [pic] [pic]Area under curve between -2 and -8
Factoring out the same width we get
The actual area
When we use this type of sum to approximate the area under curve, the sum is called a Riemann Sum. Since we are using rectangles, it is also called a rectangular approximation method or RAM for short. In this particular RAM, we used the left endpoints of the rectangle as a basis for obtaining the height of the rectangle. Using this method is abbreviated as LRAM.
Let’s examine the same problem, but let’s use right endpoints instead or RRAM.
Let’s calculate the RRAM manually.
Let’s examine a midpoint Rectangular Approximation method or MRAM
Let’s go to
Medial Summary:
If [pic]is non negative on [pic], we interpret [pic]as the area bound by [pic], below by the [pic], and vertically by the lines [pic]and [pic]. The value of the definite integral is then approximated by dividing the area into [pic]strips, approximating the area of each strip by a rectangle or other geometric figure, then summing these approximations.
We may approximate [pic] by any one of the following sums:
1) Left Sum : [pic], using the value of [pic]at the left endpoint of each interval
2) Right Sum : [pic], using the value of [pic]at the right endpoint of each interval
3) Midpoint Sum: [pic]
Any of these sums is called a Riemann Sum
Example #1:
Approximate [pic]by using four subintervals and calculating right sum
Sample Test Question #1:
1) Use [pic]to find the approximate area of the shaded region
A) 9 B) 19 C) 36 D) 38 E) 54
Example #2:
Use L(4) to approximate the area of the region bounded by [pic], the x-axis, [pic]and [pic]
Example #3:
Using right endpoints, approximate the area between the curve [pic] and the [pic]axis between [pic]and [pic]using 6 rectangles.
Example #4:
Using right endpoints, approximate the area between the curve [pic] and the [pic]axis between [pic]and [pic]using 8 rectangles
Example #5:
Using right endpoints, approximate the area between the curve [pic] and the [pic]axis between [pic]and [pic]using 5 rectangles
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[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
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