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0223 - Math 31A 2010.02.23 MATH 31A DISCUSSION JED YANG 1...

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Math 31A 2010.02.23 MATH 31A DISCUSSION JED YANG 1. Basic Integration 1.1. Exercise 5.1.71. Describe the area represented by the limit lim N →∞ 3 N N summationdisplay j =1 parenleftbigg 2 + 3 j N parenrightbigg 4 . Solution. Consider the area under the curve f ( x ) from x = a to x = b . We divide the interval b a into N equal slots, so each slot has width Δ x = b - a N . The right end points of the j th rectangle is a + j Δ x . The value of the function at the right end point of the j th rectangle is therefore f ( a + j Δ x ). If we use these as the height of the rectangles, then the area of the j th rectangle is Δ x · f ( a + j Δ x ). We sum this up to get N j =1 Δ x · f ( a + j Δ x ). We can factor out the width of the rectangles as they are all the same. Thus we get Δ x N j =1 f ( a + j Δ x ). If we let N goes to infinity, the approximation becomes finer and finer, and approaches the area under the curve. Here a = 2, b a = 3 so b = 5, and f ( x ) = x 4 . Thus the limit represents the area between the graph of x 4 and the x
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