Chapter 5.2, Problem 32E

### 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

# Approximating the Area of a Plane Region In Exercises 29-34, use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. g ( x ) = x 2 + 1 , [ 1 ,   3 ] ,   8   rectangles

To determine

To calculate: The approximate area of the region between the provided function g(x)=x2+1 in the interval [1,3] and xaxis.

Explanation

Given: The provided function is:

g(x)=x2+1 in the interval [1,3].

And the number of rectangles to be used for calculation of area is 8.

Formula used: The sum of first n natural is given by the formula:

âˆ‘i=1ni=n(n+1)2

The sum of the squares of first n natural numbers is given by the formula:

âˆ‘i=1ni2=n(n+1)(2n+1)6

The sum of a constant n times is written as:

âˆ‘i=1nc=nc

And (a+b)2=a2+b2+2ab

Calculation:

The right endpoints of the eight rectangles are:

1+(3âˆ’18)i=1+i4

Where i=â€‰1,2,3,4,5,6,7,8

The width of each rectangle

Width=3âˆ’18=14

And, height of each rectangle can be obtained from the value of the provided function at the right endpoints of each rectangle.

Approximate area between the provided function and the x-axis is equal to the sum of the eight

rectangles and the sum of the areas of the eight rectangles is:

âˆ‘i=18g(1+i4)(14)

Put value of g(1+i4),

âˆ‘i=18g(1+i4)(14)=âˆ‘i=18[(1+i4)2+1](14)

Split the summation into parts to use the summation formula:

âˆ‘i=1814(1+i4)2+âˆ‘i=1814

Factor out 14 from the first summation

14âˆ‘i=18(1+i4)2+âˆ‘i=1814

Use the formula:

(a+b)2=a2+b2+2ab

So,

14âˆ‘i=18(1+i4)2+âˆ‘i=1814=14âˆ‘i=18(i216+1+i2)+8(14)=14âˆ‘i=18i216+14âˆ‘i=181+14âˆ‘i=18i2+2

Apply the formulas:

âˆ‘i=1ni=n(n+1)2, âˆ‘i=1nc=nc, and âˆ‘i=1ni2=n(n+1)(2n+1)6.

So,

âˆ‘i=18i216+14âˆ‘i=181+14âˆ‘i=18i2+2=164(8Ã—9(16+1)6)+14(8)+18(8Ã—92)+2=15348+2+92+2=11.6875

The left endpoints of the eight rectangles are:

1+14(iâˆ’1)=i+34

Where i=â€‰1,2,3,4,5,6,7,8

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