4.2 Area

Sigma Notation

Summation Examples Example: Example: Example: Example:

More Summation Examples

Theorem 4.2 Summation Rules

Theorem 4.2 Summation Rules

Examples Example 2 Evaluate the summation Solution

Examples Example 3 Compute Solution

Examples Example 4 Evaluate the summation for n = 100 and 10000 Note that we change (shift) the upper and lower bound Solution For n = 100 For n = 10000

Summation and Limits Example 5 Find the limit for

Continued…

Area 2

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

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

Continued… The average of the lower and upper approximations is L U L 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(mi) and that of circumscribed rectangle is f(Mi).

Upper and Lower Sums The i-th regional area Ai 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

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

Exact Area Using the Limit

Definition of the Area of a Region in the Plane

In General - Finding Area Using the Limit b height x base Area = Or, xi , the i-th right endpoint

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

Regular Right-Endpoint Formula

Continued

Homework Pg. 267 1, 7, 11, 15, 21, 31, 33, 41, 23-29 odd, 39, 43