Chapter 5.2, Problem 43E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
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 subintervalsFunctionInterval f ( x ) = 5 x 2 [0,1]

To determine

To calculate: The upper and lower sums for the region bounded by the graph of the provided function f(x)=5x2 in the interval [0,1] and x -axis.

Explanation

Given: The provided function is f(x)=5x2 in the interval [0,1].

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

Formula used: The sum of first n natural is given by the formula: âˆ‘i=1ni=n(n+1)2

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

Lower sum is calculated by formula: s=âˆ‘i=1nf(mi)Î”x, Where mi are the left endpoints.

And the upper sum is calculated by: S=âˆ‘i=1nf(Mi)Î”x, where Mi are the right endpoints.

Calculation: To begin, partition the interval [0,1] into n subintervals, each of width

Î”x=1âˆ’0n=1n

Left endpoints (mi),

mi=0+(iâˆ’1)(1n)=(iâˆ’1n)

i=â€‰1,2,3,.......,n The lower sum (s) can be calculated by the formula:

s=âˆ‘i=1nf(mi)Î”x,

Where, mi are the left end points.

By the left endpoints, the lower sum is: s=âˆ‘i=1nf(iâˆ’1n)(1n)

Put value of f(iâˆ’1n),

Factor out 5n3 from the sum,

âˆ‘i=1n5(i2+1âˆ’2in2)(1n)=5n3âˆ‘i=1n(i2âˆ’2i+1)

Split the summation into parts to use summation formulas:

5n3âˆ‘i=1n(i2âˆ’2i+1)=5n3âˆ‘i=1ni2âˆ’5n3âˆ‘i=1n2i+5n3âˆ‘i=1n1

Apply the formulas: âˆ‘i=1ni=n(n+1)2â€‰â€‰â€‰â€‰,â€‰â€‰âˆ‘i=1nc=ncâ€‰,andâ€‰â€‰âˆ‘i=1ni2=n(n+1)(2n+1)6

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