   Chapter 4.2, Problem 31E

Chapter
Section
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 ) = 2 x 2 − x − 1 , [ 2 , 5 ] , 6   rectangles

To determine

To calculate: Approximate area of the region between the function g(x)=2x2x1 and xaxis.

Explanation

Given: g(x)=2x2x1 in the interval [2,5].

Number of rectangles to be used for calculation of area is 6.

Formula used: Formula for the sum of first n natural is:

i=1ni=n(n+1)2

Formula for the sum of the squares of first n natural numbers is:

i=1ni2=n(n+1)(2n+1)6

The sum of a constant n times is written as:

i=1nc=nc

Also, (a+b)2=a2+b2+2ab.

Calculation: The right endpoints of the six rectangles are:

2+i2

Where i=1,2,3,4,5,6.

The width of each rectangle:

Width=526=12

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 sixrectangles and the sum of the areas of the six rectangles is:

i=16f(2+i2)(12)

Put value of f(2+i2),

i=16f(2+i2)(12)=i=16[2(2+i2)2(2+i2)1](12)

Split the summation into parts to use the summation formula:

i=16[2(2+i2)2(2+i2)1](12)=i=16(2+i2)2i=16(1+i4)i=1612

Use the formula:

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

Hence,

i=16(2+i2)2i=16(1+i4)i=1612=i=16(4+i24+2i)i=161i=16i4i=1612=i=16i24+i=1674i+i=1652

Apply the formulae:

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

So,

i=16i24+i=1674i+i=1652=14(6×7×136)+74(6×72)+6(52)=914+1474+604=74.5

The left endpoints of the four rectangles are:

2+12(i1)=i+32

Where i=1,2,3,4,5,6.

The width of each rectangle is 12 and height of each rectangle can be obtained from the value of the provided function at the left endpoints of each rectangle

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