   Chapter 4.2, Problem 42E

Chapter
Section
Textbook Problem

Finding Upper and Lower Sums for a Region In Exercises 41-44, find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals.Function Interval f ( x ) = 6 − 2 x ⌈ 1 , 2 ⌉

To determine

To calculate: The upper and lower sums for the region bounded by the graph of f(x)=62x in the interval [1,2] and x -axis.

Explanation

Given: f(x)=62x in the interval [1,2]

Number of subintervals to be used in the calculation of area is n.

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

The sum of a constant n times is written as:i=1nc=nc

Formula to calculating Lower sum:s=i=1nf(mi)Δx,

Where, mi are the left endpoints.

Formula for calculating upper sum:

S=i=1nf(Mi)Δx, where Mi are the right endpoints.

Calculation: Partition the interval [1,2] into n subintervals, each of width:

Δx=21n=1n

Left endpoints (mi),

mi=1+(i1)(1n)=(i+n1n)

Formula to calculate the lower sum (s):

s=i=1nf(mi)Δx,

Where, mi are the left endpoints.

Lower sum by the left endpointsis:

s=i=1nf(i+n1n)(1n)

Use value of f(i+n1n),

s=i=1n(62(i+n1n))(1n)=i=1n(4n2i+2n2)

Factor out 1n2 from the sum,

i=1n(4n2i+2n2)=1n2i=1n(4n2i+2)

Split the summation into parts to use summation formulas:

1n2i=1n(4n2i+2)=1n2i=1n4n1n2i=1n2i+1n2i=1n2

Using the formulas:

i=1ni=n(n+1)2,andi=1nc=nc

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