BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5.2, Problem 15E
To determine

To Calculate: The right Riemann sums Rn for the integral 0πsinxdx for n=5,10,50,and100 using the calculator.

To Find: The values of the numbers appear to be approaching.

Expert Solution

Answer to Problem 15E

The value of right Riemann sums for the integral 0πsinxdx for n=5,10,50,and100 is tabulated in table 1.

The value in the Table 1 is approaching to a value of 2.

Explanation of Solution

Given:

The integral function as 0πsinxdx

Number of rectangles n=5,10,50and100.

Calculation:

Show the Equation of the integral as follows:

0πsinxdx (1)

Consider the value of the function f(x)=sinx (2)

For n=5

Find the width (Δx) using the relation:

Δx=ban (3)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

The limits of the integral 0πsinxdx, a=0 and b=π.

Substitute π for b, 5 for n, and 0 for a in Equation (3).

Δx=π05=π5

The right endpoints are x1=π5, x2=2π5, x3=3π5, x4=4π5 and x5=π.

The expression to find the right Riemann sum Rn as shown below:

Rn=i=1nf(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(xn)Δx (4)

Here, the right endpoint height of first rectangle is f(x1), the width is Δx, height of right endpoint of second rectangle is f(x2), and left endpoint height of nth rectangle is f(xn).

Calculate the value of f(x1) using the Equation (2).

Substitute π5 for x1 in Equation (2).

f(x1)=sinx1=sin(π5)=0.0109

Calculate the value of f(x2) using the Equation (2).

Substitute 2π5 for x2 in Equation (2).

f(x2)=sinx2=sin(2π5)=0.02193

Similarly calculate the value of f(x) for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute π5 for Δx, 5 for n, and values of right end points in Equation (4).

Rn=i=15f(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(x5)Δx=f(π5)×π5+f(2π5)×π5+...+f(π)×π5=1.933766

The value of right Riemann Sum for n=5 is 1.933766_.

For n=10

Find the width (Δx) using the Equation (3):

Substitute π for b , 10 for n, and 0 for a in Equation (3).

Δx=π010=π10

The right endpoints are x1=π10, x2=2π10, x3=3π10x10=π.

Calculate the value of f(x1) using the Equation (2).

Substitute π10 for x1 in Equation (2).

f(x1)=sinx1=sin(π10)=5.483×103

Calculate the value of f(x2) using the Equation (2).

Substitute 2π10 for x2 in Equation (2).

f(x2)=sinx2=sin(2π10)=0.01096

Similarly calculate the value of f(x) for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute π10 for Δx, 10 for n, and values of right end points in Equation (4).

R10=i=110f(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(x10)Δx=f(π10)×π10+f(2π10)×π10+...+f(π)×π10=1.983524

The value of right Riemann Sum for n=10 is 1.983524_.

For n=50

Find the width (Δx) using the Equation (3):

Substitute π for b , 50 for n, and 0 for a in Equation (3).

Δx=π050=π50

The right endpoints are x1=π50, x2=2π50, x3=3π50x50=π.

Calculate the value of f(x1) using the Equation (2).

Substitute π50 for x1 in Equation (2).

f(x1)=sinx1=sin(π50)=1.0966×103

Calculate the value of f(x2) using the Equation (2).

Substitute 2π50 for x2 in Equation (2).

f(x2)=sinx2=sin(2π50)=2.1932×103

Similarly calculate the value of f(x) for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute π50 for Δx, 50 for n, and values of right end points in Equation (4).

R50=i=150f(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(x50)Δx=f(π50)×π50+f(2π50)×π50+...+f(π)×π50=1.999342

The value of right Riemann Sum for n=50 is 1.999342_.

For n=100

Find the width (Δx) using the Equation (3):

Substitute π for b , 100 for n, and 0 for a in Equation (3).

Δx=π0100=π100

The right endpoints are x1=π100, x2=2π100, x3=3π100x100=π.

The expression to find the right Riemann sum Rn as shown below:

Rn=i=1nf(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(xn)Δx (5)

Here, the right endpoint height of first rectangle is f(x1), the width is Δx, height of right endpoint of second rectangle is f(x2), and left endpoint height of nth rectangle is f(xn).

Calculate the value of f(x1) using the Equation (2).

Substitute π100 for x1 in Equation (2).

f(x1)=sinx1=sin(π100)=5.483×104

Calculate the value of f(x2) using the Equation (2).

Substitute 2π100 for x2 in Equation (2).

f(x2)=sinx2=sin(2π100)=1.0966×103

Similarly calculate the value of f(x) for remaining values of x.

Calculate the right Riemann sum using calculator.

Substitute π100 for Δx, 100 for n, and values of right end points in Equation (5).

R100=i=1100f(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(x50)Δx=f(π100)×π100+f(2π100)×π100+...+f(π)×π100=1.999836

The value of right Riemann Sum for n=100 is 1.999836_.

Tabulate the values of Riemann sum for n=5,10,50and100 as shown in table 1.

nRn
51.933766
101.983524
501.999342
1001.999836

Table 1

Refer to Table 1.

The value of the Riemann sum is approaching to 2.

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