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Stewart6e5_1

Stewart6e5_1 - MATH 191 Sections 1 and 3 Calculus I Fall...

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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 5.1: Areas! Today we’ll work on perhaps the funnest topic of the semester. We all know how to find the area of nice rectilinear figures like squares and triangles, right? What if we’re asked to find the area of a curvilinear figure, like the region bounded above by the parabola f ( x ) = x 2 , over the interval [0 , 2]? The best way to get a grip on this problem is to begin by the area, dividing it up into rectangles, which we know how to handle! Draw a figure below in which you use 4 rectangles of equal width to over estimate the area we seek: Notice that the rectangles we chose make use of the value of the function f ( x ) = x 2 at the endpoint, x i , of the i th subinterval in order to determine the rectangle’s height. It’s reasonable to call this estimate R 4 , where the 4 refers to the number of rectangles we used. What estimate do we obtain for the region’s area? ( I.e. , what’s R 4 ?) How can we get a better overestimate for the actual area? What happens if we do this? What if we want an...
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Stewart6e5_1 - MATH 191 Sections 1 and 3 Calculus I Fall...

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