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Unformatted text preview: 22. Areas and Sums P. K. Lamm 11/13/11 (23:18) p. 1 / 15 Lecture Notes: 22. Areas and Sums These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Approximating Area Consider the problem of approximating the area enclosed by the curve y = x 2 + 1 , the xaxis, and the lines x = 0 and x = 2. In what follows we will let f ( x ) = x 2 + 1. Example 1.1 – Right endpoints: We will initially approximate this area using 4 equalwidth rectangles with heights of the rectangles determined by the height of the curve y = x 2 + 1 at the right side of each rectangle. If the rectangles are to be equalwidth, their width is given by Δ x = (2 0) / 4 = . 5 , since the interval [0 , 2] in x is to be divided into 4 equal parts. The right endpoints of these rectangles are the points c 1 = 0 . 5 , c 2 = 1 . , c 3 = 1 . 5 , c 4 = 2 . , where we use the subscript to designate the rectangle number; that is, the rightendpoint c 1 is for the 1st rectangle, c 2 is for the 2nd rectangle, etc.. We could have used different variables like a , b , 1 22. Areas and Sums P. K. Lamm 11/13/11 (23:18) p. 2 / 15 c , ... for these endpoints, but using a subscript allows for a more compact way of associating the endpoint with the appropriate rectangle. It also allows us to refer to all four endpoints using the shorthand notation c k , for k = 1 , 2 , 3 , 4 , and, since there is a pattern involved in finding the values of each c k , we obtain the formula c k = k Δ x, for k = 1 , 2 , 3 , 4 . The area of the first rectangle is height × base = f (0 . 5) × . 5 = (0 . 5 2 + 1)( . 5) = (1 . 25)( . 5) , since the height of the rectangle is the same as the height of the curve at the first right endpoint c 1 = . 5. Similarly, the area of the second rectangle is height × base = f (1 . 0) × . 5 = (1 . 2 + 1)( . 5) = (2 . 00)( . 5) . We can summarize what we’ve found so far, and find an estimate for the approximate area using four rectangles as follows: Area of rectangle #1 = f ( c 1 )Δ x = f (0 . 5)( . 5) = (0 . 5 2 + 1)( . 5) = (1 . 25)( . 5) + Area of rectangle #2 = f ( c 2 )Δ x = f (1 . 0)( . 5) = (1 . 2 + 1)( . 5) = (2 . 00)( . 5) + Area of rectangle #3 = f ( c 3 )Δ x = f (1 . 5)( . 5) = (1 . 5 2 + 1)( . 5) = (3 . 25)( . 5) + Area of rectangle #4 = f ( c 4 )Δ x = f (2 . 0)( . 5) = (2 . 2 + 1)( . 5) = (5 . 00)( . 5) SUM of AREAS = (11 . 50)( . 5) = 5 . 75 Thus, an approximation for the area under the curve is 5.75 square units. From the picture shown above, it is clear that this approximation is greater than the area in question....
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 Fall '10
 KIHYUNHYUN
 Calculus, Riemann sum, P. K. Lamm

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