   Chapter 6.3, Problem 47E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Consumer Trends The rate of change in the number of subscribers S to a newly introduced magazine is modeled by d S d t =   1000 t 2 e − t ,   0   ≤   t   ≤   6 where t is the time in years. Use Simpson's Rule with n   = 12 to estimate the total increase in the number of subscribers during the first 6 years.

To determine

To calculate: The total increase in the number of subscribers during the first 6 years by the Simpson’s Rule with n=12 if the rate of change in the number of subscriber is dSdt=1000t2et where 0t6.

Explanation

Given Information:

The rate of change in the number of subscriber is,

dSdt=1000t2et

Where, 0t6.

Formula used:

Simpson’s Rule:

If f is continuous on [a,b] and n is an even integer, then

abf(x)dx,n=(ba3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)++4f(xn1)+f(xn)]

Calculation:

Consider the rate of change in the number of subscriber is,

dSdt=1000t2et

So, the definite integral is 061000t2etdt.

Here,

a=0b=6

When n=12, the width of each subinterval is,

6012=12

And the end points of subintervals are,

x0=0x1=12x2=12+12=1

And,

x3=1+12=32x4=32+12=2

And,

x5=2+12=52x6=52+12=3

And,

x7=62+12=72x8=72+12=4

And,

x9=4+12=92x10=92+12=5

And,

x11=102+12=112x12=112+12=6

Now, use Simpson’s rule,

061000t2e

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