Calculus II Notes 5.1 - Calculus II-Stewart Dr. Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 5.1 5.1 Areas and Distances Integral calculus arose from the need to calculate areas and volumes of complex shapes. The Area Problem Before the Fundamental Theorem of Calculus was discovered, the standard method of calculating an area was to approximate it using rectangles. Example A We approximate the area between the x -axis and the graph of y = x 2 on the interval [0,1] using rectangles of width 1/4. First we inscribe the rectangles under the curve to get a lower estimate, and then we inscribe the rectangles above the curve to get an upper estimate. Lower Estimate: A l = 1 4 0 ( ) 2 + 1 4 ( ) 2 + 2 4 ( ) 2 + 3 4 ( ) 2 [ ] = 1 4 1 16 + 4 16 + 9 16 [ ] = 1 4 14 16 [ ] = 7 32 Upper Estimate: A u = 1 4 1 4 ( ) 2 + 2 4 ( ) 2 + 3 4 ( ) 2 + 4 4 ( ) 2 [ ] = 1 4 1 16 + 4 16 + 9 16 + 1 [ ] = 1 4 30 16 [ ] = 15 32 We can get better upper and lower estimates by using narrower rectangles. (See the textbook.) Finding the Exact Area In many cases we can find the exact area using algebra and a little elementary analysis. Suppose we wish to calculate the area under a positive curve and the x -axis on the closed interval I = [ a , b ] . For the sake of simplicity, we use n rectangles each of the
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Calculus II Notes 5.1 - Calculus II-Stewart Dr. Berg Spring...

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