11.5 Area 2014
After this lesson, you should be able to: Use sigma notation to write and evaluate a sum. Understand the concept of area. Approximate the area of a plane region. Find the area of a plane region using limits.
Sigma Notation
Summation Examples Example:
Example 1 More Summation Examples
Summation Rules
Example 1 Evaluate the summation Solution Examples
Example 2 Compute Solution Examples
Example 3 Evaluate the summation for n = 100 and Solution Note that we change (shift) the upper and lower bound For n = 100For n = Examples
Summation and Limits Example 4 Find the limit for
Continued…
Review Example : Evaluate the following limit:
Area 2 Area of the region bounded by and the lines x=2 and y=0 ?
Lower Approximation Using 4 inscribed rectangles of equal width Lower approximation = (sum of the rectangles) 2
Using 4 circumscribed rectangles of equal width Upper approximation = (sum of the rectangles) 2 Upper Approximation
Continued… LU LAU A The average of the lower and upper approximations is A is approximately
Upper and Lower Sums The procedure we just used can be generalized to the methodology to calculate the area of a plane region. We begin with subdividing the interval [ a, b ] into n subintervals, each of equal width x = ( b – a )/ n. The endpoints of the intervals are
Upper and Lower Sums Because the function f(x) is continuous, the Extreme Value Theorem guarantees the existence of a minimum and a maximum value of f(x) in each subinterval. We know that the height of the i -th inscribed rectangle is f(m i ) and that of circumscribed rectangle is f(M i ).
Upper and Lower Sums The i-th regional area A i is bounded by the inscribed and circumscribed rectangles. We know that the relationship among the Lower Sum, area of the region, and the Upper Sum is
The procedure we just used can be generalized to the methodology to calculate the area of a plane region. We begin with subdividing the interval [ a, b ] into n subintervals, each of equal width x = ( b – a )/ n.
Theorem 4.3 Limits of the Upper and Lower Sums
ab Area = heightxbase In General - Finding Area Using the Limit Or, x i, the i -th right endpoint
2 length =2 – 0 =2 n = # of rectangles Exact Area Using the Limit
Definition of the Area of a Region in the Plane
Regular Right-Endpoint Formula RR-EF Example 6 Find the area under the graph of 15 A =
Regular Right-Endpoint Formula
Continued
Regular Right-Endpoint Formula RR-EF Example 7 Find the area bounded by the graph of f(x), the x-axis, the y-axis, and x = 3.
Regular Right-Endpoint Formula RR-EF Example 8 Find the area bounded by the graph of f(x), and the x-axis on the given interval
Homework Day 1: Section 11.5 pg odd, odd Day 2: Section 11.5 pg even Day 3: Ch. 11 Review pg odd Day 4: Ch. 11 Practice Test Ch. 11 Test Monday 5/11
HWQ Find the area between the graph of f(x) and the x-axis on the given interval: