Area Section 4.2.

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Presentation transcript:

Area Section 4.2

Summation Sigma is used to denote summation. MAT 224 SPRING 2007 Summation Sigma is used to denote summation. The sum of n terms a1, a2, a3, …, an is expressed as i is called the ______________________________________ 1 is the ____________________________; n is the ___________________________ . ai is the ith term of the sum.

Summation Examples Example: Example: Example:

Summation Example:

Summation Rules

Summation Rules

Examples

Area 2

Consider the region bounded by the graphs of The area can be approximated by two sets of rectangles—one set inscribed within the region and the other set circumscribed over the region. f(x)=x2 Circumscribed rectangles Upper Sum f(x)=x2 Inscribed rectangles Lower Sum f(x)=x2 The actual area lies between the lower and upper sums.

Lower Approximation Find the sum of the areas of the inscribed rectangles. f(x)=x2 Inscribed rectangles Lower Sum

Upper Approximation Find the sum of the areas of the circumscribed rectangles. f(x)=x2 Circumscribed rectangles Upper Sum

Continued… The actual area lies between the lower and upper sums. L A Thus, the area bounded by the graphs of is

Example Find the lower and upper approximations of the area of the region lying between the graph of and the x-axis between x = 0 and x = 2. Use 4 rectangles. 1) Lower Sum

Example 2) Upper Sum

The limit In fact, the exact area can be found by: Area under curve = **Notice: The smaller the intervals (the greater the number of rectangles), the closer the approximate area is to the actual area. In fact, the exact area can be found by: Area under curve = Lower Sum Upper Sum

As n increases without bound As you increase the number of rectangles, the approximation tends to become better because the amount of ‘missed area’ decreases. Check this out: http://xanadu.math.utah.edu/java/ApproxArea.html

Homework Section 4.2 page 267 # 1 – 7 odd, 15, 23, 27, 29