   Chapter 4.2, Problem 30E

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. f ( x ) = 9 − x , [ 2 , 4 ] , 6   rectangles

To determine

To calculate: Approximate area of the region between the function f(x)=9x in the interval [2,4] and x-axis.

Explanation

Given: f(x)=9x in the interval [2,4].

Number of rectangles to be used for calculation of areais 6.

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

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

Sum of a constant n times is written as:

i=1nc=nc

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

Calculation: The right endpoints of the six rectangles are:

2+i3

where, i=1,2,3,4,5,6

The width of each rectangle,

Width=426=13

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+i3)(13)

Put value of f(2+i3),

i=16f(2+i3)(13)=i=16[7i3](13)

Split the summation in two parts to use the summation formula:

i=16[7i3](13)=i=1673i=16i9

Factor out 19 from the second sum,

i=1673i=16i9=i=167319i=16i

Apply the formulas:

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

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