4.2 Area

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

Theorem 4.2 Summation Rules

Example 2 Evaluate the summation Solution Examples

Example 3 Compute Solution Examples

Example 4 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 5 Find the limit for

Continued…

Area 2

Lower Approximation Using 4 inscribed rectangles of equal width Lower approximation = (sum of the rectangles) 2 The total number of inscribed rectangles

Using 4 circumscribed rectangles of equal width Upper approximation = (sum of the rectangles) 2 Upper Approximation The total number of circumscribed rectangles

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

Theorem 4.3 Limits of the Upper and Lower Sums

2 length =2 – 0 =2 n = # of rectangles Exact Area Using the Limit

Definition of the Area of a Region in the Plane

ab Area = heightxbase In General - Finding Area Using the Limit Or, x i, the i -th right endpoint

Regular Right-Endpoint Formula RR-EF intervals are regular in length squaring from right endpt of rect. Example 6 Find the area under the graph of 15 a = 1 b = 5 A =

Regular Right-Endpoint Formula

Continued

Homework Section 4.2 page 261 #1-7 odd, 15, 17, 29, 31, 33, 39, 41, 47, 49, 51