Please help with this wrong answer? (Circled in red) EXAMPLE 2For the region under f(x) = 5x2 on [0, 21, show that the sum of the areas ofthe upper approximating rectangle approachesthat is40lim Rnn 00SOLUTIONRn is the sum of the areas of the n rectangles in the figure. Each rectangle haswidthand the heights are the values of the function f(x) = 5x2 at the points46...,"; that is, the heights aren'2nThus,2.2.5Rn =++... +n410· (12 + 22 + 32 ++ n?)n240(12 + 22 + 32 + ... + n?).n°„3Here we need the formula for the sum of the squares of the first n positive integers:12 + 22 + 32 +... + n2 - n(n + 1)(2n + 1)Perhaps you have seen this formula before. Putting this formula into our expression for Ro.we get40Rn =n340(n +1)(2n + 1)6n2n(n + 1)(2n + 1)Thus we have40n2 + 60n + 20lim Rn = limn 003n?20 2+= limn- 00n2n +)12n + 1in320 +0+0(1 + +)(2 + +)= lim32040•1·2 =

Name: Please help with this wrong answer? (Circled in red) EXAMPLE 2For the
Uploaded: 2024-03-20T07:07:37.000Z
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Description: Please help with this wrong answer? (Circled in red) EXAMPLE 2For the region under f(x) = 5x2 on [0, 21, show that the sum of the areas ofthe upper approximating rectangle approachesthat is40lim Rnn 00SOLUTIONRn is the sum of the areas of the n rect