# 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 5E

a.

To determine

## To estimate the right endpoints.

Expert Solution

The value of right endpoints is 4.

### Explanation of Solution

Given information:

The equation is

08f(x)dx with n=4

Calculation:

The right endpoints can be estimate as

08f(x)dx

The endpoints of the 4 subintervals are [0,2],[2,4],[4,6],[6,8]

So, the right endpoints are 2,4,6,8

The width of the subintervals is

Δx=banΔx=804          [since, b=8,a=0,n=4]Δx=84Δx=2              [divede numerator and denominator by 4]

From the given graph in the question we can observed that

f(2)=1f(4)=2f(6)=2f(8)=1

Therefore,

The right endpoint rule gives

08f(x)dx=Δx[f(2)+f(4)+f(6)+f(8)]

Put the values of Δx=2,f(2)=1,f(4)=2,f(6)=2 and f(8)=1 on the above equation

08f(x)dx=Δx[f(2)+f(4)+f(6)+f(8)]                 =2(1+2+(2)+1)                 =2(1+22+1)                 =2(2)                 =4

Hence,

The value of right endpoints is 4.

b.

To determine

### To estimate the left endpoints.

Expert Solution

The value of left endpoints is 6.

### Explanation of Solution

Given information:

The equation is

08f(x)dx with n=4

Calculation:

The left endpoints can be estimate as

08f(x)dx

The endpoints of the 4 subintervals are [0,2],[2,4],[4,6],[6,8]

So, the left endpoints are 0,2,4,6

The width of the subintervals is

Δx=banΔx=804          [since, b=8,a=0,n=4]Δx=84Δx=2              [divede numerator and denominator by 4]

From the given graph in the question we can observed that

f(0)=2f(2)=1f(4)=2f(6)=2

Therefore,

The right endpoint rule gives

08f(x)dx=Δx[f(0)+f(2)+f(4)+f(6)]

Put the values of Δx=2,f(0)=2,f(2)=1,f(4)=2 and f(6)=2 on the above equation

08f(x)dx=Δx[f(0)+f(2)+f(4)+f(6)]                 =2(2+1+2+(2))                 =2(2+1+22)                 =2(3)                 =6

Hence,

The value of left endpoints is 6.

c.

To determine

### To estimate the midpoints.

Expert Solution

The value of midpoints is 10.

### Explanation of Solution

Given information:

The equation is

08f(x)dx with n=4

Calculation:

The midpoints can be estimate as

08f(x)dx

The endpoints of the 4 subintervals are [0,2],[2,4],[4,6],[6,8]

So, the midpoints are 1,3,,5,7

The width of the subintervals is

Δx=banΔx=804          [since, b=8,a=0,n=4]Δx=84Δx=2              [divede numerator and denominator by 4]

From the given graph in the question we can observed that

f(1)=3f(3)=2f(5)=1f(7)=1

Therefore,

The midpoint rule gives

08f(x)dx=Δx[f(1)+f(3)+f(5)+f(7)]

Put the values of Δx=2,f(1)=3,f(3)=2,f(5)=1 and f(7)=1 on the above equation

08f(x)dx=Δx[f(1)+f(3)+f(5)+f(7)]                 =2(3+2+1+(1))                 =2(3+2+11)                 =2(5)                 =10

Hence,

The value of midpoints is 10.

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