   Chapter 4.3, Problem 45E

Chapter
Section
Textbook Problem

Estimating a Definite Integral Use the table of values to find lower and upper estimates of ∫ 0 10 f ( x ) d x Assume that f is a decreasing function. x 0 2 4 6 8 10 F(x) 32 24 12 -4 -20 -36

To determine

To calculate: The value of the provided integral 010f(x)dx using left and right endpoints assuming the function is a decreasing function and the values of the function are

 x 0 2 4 6 8 10 f(x) 32 24 12 −4 −20 −36
Explanation

Given:

The integral to be calculated is 010f(x)dx

And the table is provided, which is given below

 x 0 2 4 6 8 10 f(x) 32 24 12 −4 −20 −36

Formula used:

The definite integral of f(x) from a to b is defined as:

abf(x)dx=limni=1nf(ci)Δxi

Here a is the lower limit of integration and b is the upper limit of integration.

Calculation:

The formula for integration using left endpoints is

abf(x)dx=limni=1nf(ci)Δxi

Here ci are the left endpoints.

From the provided table

The width of each subinterval, Δx=2 and n=5

Find Left endpoints

ci=a+bi1n=0+10(i16)=53(i1)

So,

limn i=1nf(ci)Δxi =limn(2f(0)+2f(2)+2f(4)+2f(6)+2f(8))

Use the table for values of function,

 x 0 2 4 6 8 10 f(x)

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