   Chapter 4.3, Problem 3E

Chapter
Section
Textbook Problem

Evaluating a Limit In Exercises 3 and 4, use Example 1 as a model to evaluate the limit lim n → ∞ ∑ i = 1 n f ( c i ) Δ x i over the region bounded by the graphs of the equations. f ( x ) = x , y = 0 , x = 0 , x = 3 (Hint: Let c i = 3 i 2 n 2 )

To determine

To calculate: Riemann’s limit over the region bounded by the graphs of the equations f(x)=xy=0,x=0,x=3.

Explanation

Given: f(x)=x, y=0,x=0,x=3

Limit to evaluate is:limni=1nf(ci)Δxi

Formula used:

Formula for the sum of first n natural numbers:i=1ni=n(n+1)2

Formula for the sum of the squares of first n natural numbers:i=1ni2=n(n+1)(2n+1)6

Calculation: Givenlimit to evaluate: limni=1nf(ci)Δxi

where ci is the right endpoints of the partition and Δxi is the width of the ith interval.

Assume that ci=3i2n2.

Therefore, the width of the ith interval is given as:

Δxi=3i2n23(i1)2n2=3i23(i2+12i)n2=6i3n2

Limit is:

limni=1nf(ci)Δxi=limni=1nf(3i2n2)Δxi

Use value of f(3i2n2) and Δxi in above equation:

limni=1nf(3i2n2)Δxi=limni=1n3i2n2(6i3n2)=limni=1n3ni(6i3n2)

Factor out 3n3<

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