Solve the image exercise yEXAMPLE 2For the region under f(x) = 4x² on [0, 2], show that the sum of the areas of the upper approximating(2, 16)rectangle approaches ,that is32lim Rn =3R, is the sum of the areas of the n rectangles in the figure. Each rectangle has width2and the heightsSOLUTIONare the values of the function f(x) = 4x2 at the points2n-; that is, the heights aren' n' n|. Thus,2.+=88X : (12 + 22 + 32 ++ n?)(12 + 22 + 32 ++ n?).1/n2Video ExampleHere we need the formula for the sum of the squares of the first n positive integers:+ n? = n(n + 1)(2n + 1)6.12 + 22 + 32 +Perhaps you have seen this formula before. Putting this formula into our expression for Rp, we getn(n + 1)(2n + 1)R, =Thus we havelim Rn = limn + 12n + 1= limlim•1:2 =

Answered: Image /qna-images/answer/269ff221-aef9-471b-9722-2dbc89827bef.jpg

(2, 16) rectangle approaches 4, that is lim R, = SOLUTION R, is the sum of the areas of the n rectangles in the figure. Each rectangle has width and the heights are the values of the function f(x) = 4x at the points that is, the heights are ... Thus, ..... .. +2. (20) + 8 n? x (1² + 2² + 32 + ... + n²) (1² + 22 + 32 + ... + n?). 1/n 2 Video Example Here 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) 6. Perhaps you have seen this formula before. Putting this formula into our expression for Rn, we get n(n + 1)(2n + 1) R, = 6. Thus we have lim Re = lim n-00 lim (1 + X(2 + ÷) - lim •1.2 =

(2, 16) rectangle approaches 4, that is lim R, = SOLUTION R, is the sum of the areas of the n rectangles in the figure. Each rectangle has width and the heights are the values of the function f(x) = 4x at the points that is, the heights are ... Thus, ..... .. +2. (20) + 8 n? x (1² + 2² + 32 + ... + n²) (1² + 22 + 32 + ... + n?). 1/n 2 Video Example Here 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) 6. Perhaps you have seen this formula before. Putting this formula into our expression for Rn, we get n(n + 1)(2n + 1) R, = 6. Thus we have lim Re = lim n-00 lim (1 + X(2 + ÷) - lim •1.2 =

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

ChapterA: Appendix

SectionA.2: Geometric Constructions

Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...

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