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 5E

a.

To determine

To estimate the right endpoints.

Expert Solution

Answer to Problem 5E

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

Answer to Problem 5E

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

Answer to Problem 5E

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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