# To estimate the right endpoints.

### Single Variable Calculus: Concepts...

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

### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter 5.2, Problem 6E

a.

To determine

## To estimate the right endpoints.

Expert Solution

The value of right endpoints is 0.5 .

### Explanation of Solution

Given information:

The equation is

33g(x)dx with n=6

Calculation:

The right endpoints can be estimate as

33g(x)dx

The endpoints of the 6 subintervals are [3,2],[2,1],[1,0],[0,1],[1,2],[2,3]

So, the right endpoints are 2,1,0,1,2,3

The width of the subintervals is

Δx=banΔx=3(3)6          [since, b=3,a=3,n=6]Δx=3+36Δx=66Δx=1              [divede numerator and denominator by 6]

From the given graph in the question we can observed that

g(2)=1g(1)=0.5g(0)=1.5g(1)=1.5g(2)=0.5g(3)=2.5

Therefore,

The right endpoint rule gives

33g(x)dx=Δx[g(2)+g(1)+g(0)+g(1)+g(2)+g(3)]

Put the values of Δx=1,g(2)=1,g(1)=0.5,g(0)=1.5,g(1)=1.5,g(2)=0.5 and g(3)=2.5 on the above equation

33g(x)dx=Δx[g(2)+g(1)+g(0)+g(1)+g(2)+g(3)]                 =1(1+(0.5)+(1.5)+(1.5)+(0.5)+2.5)                 =1(10.51.51.50.5+2.5)                 =1(3.54)                 =1(0.5)                 =0.5

Hence,

The value of right endpoints is 0.5 .

b.

To determine

### To estimate the left endpoints.

Expert Solution

The value of left endpoints is 1 .

### Explanation of Solution

Given information:

The equation is

33g(x)dx with n=6

Calculation:

The left endpoints can be estimate as

33g(x)dx

The endpoints of the 6 subintervals are [3,2],[2,1],[1,0],[0,1],[1,2],[2,3]

So, the left endpoints are 3,2,1,0,1,2

The width of the subintervals is

Δx=banΔx=3(3)6          [since, b=3,a=3,n=6]Δx=3+36Δx=66Δx=1              [divede numerator and denominator by 6]

From the given graph in the question we can observed that

g(3)=2g(2)=1g(1)=0.5g(0)=1.5g(1)=1.5g(2)=0.5

Therefore,

The right endpoint rule gives

33g(x)dx=Δx[g(3)+g(2)+g(1)+g(0)+g(1)+g(2)]

Put the values of Δx=1,g(3)=2,g(2)=1,g(1)=0.5,g(0)=1.5,g(1)=1.5 and g(2)=0.5 on the above equation

33g(x)dx=Δx[g(3)+g(2)+g(1)+g(0)+g(1)+g(2)]                 =1(2+1+(0.5)+(1.5)+(1.5)+(0.5))                 =1(2+10.51.51.50.5)                 =1(34)                 =1(1)                 =1

Hence,

The value of left endpoints is 1 .

c.

To determine

### To estimate the midpoints.

Expert Solution

The value of midpoints is 1.75 .

### Explanation of Solution

Given information:

The equation is

33g(x)dx with n=6

Calculation:

The midpoints can be estimate as

33g(x)dx

The endpoints of the 6 subintervals are [3,2],[2,1],[1,0],[0,1],[1,2],[2,3]

So, the midpoints are 2.5,1.5,0.5,0.5,1.5,2.5

The width of the subintervals is

Δx=banΔx=3(3)6          [since, b=3,a=3,n=6]Δx=3+36Δx=66Δx=1              [divede numerator and denominator by 6]

From the given graph in the question we can observed that

g(2.5)=1.5g(1.5)=0g(0.5)=1g(0.5)=1.75g(1.5)=1g(2.5)=0.5

Therefore,

The midpoint rule gives

33g(x)dx=Δx[g(2.5)+g(1.5)+g(0.5)+g(0.5)+g(1.5)+g(2.5)]

Put the values of Δx=1,g(2.5)=1.5,g(1.5)=0.5,g(0.5)=1,g(0.5)=1.75,g(1.5)=1 and g(2.5)=0.5 on the above equation

33g(x)dx=Δx[g(2.5)+g(1.5)+g(0.5)+g(0.5)+g(1.5)+g(2.5)]                 =1(1.5+0+(1)+(1.75)+(1)+0.5)                 =1(1.5+011.751+0.5)                 =1(23.75)                 =1(1.75)                 =1.75

Hence,

The value of midpoints is 1.75 .

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