# The left and right Riemann sums for the function f ( x ) = x ( x + 1 ) on interval [ 0 , 2 ] and n = 100 using the calculator. To Explain : the reason for the expression of estimate as follows: 0.8946 &lt; ∫ 0 2 x x + 1 d x &lt; 0.9081

### 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 14E
To determine

## To Calculate: The left and right Riemann sums for the function f(x)=x(x+1) on interval [0,2] and n=100 using the calculator.To Explain: the reason for the expression of estimate as follows:0.8946<∫02xx+1dx<0.9081

Expert Solution

The value of left Riemann sums for the function f(x)=1(x+1) on interval [0,2] and n=100 using the calculator is 0.89469_.

The value of right Riemann sums for the function f(x)=1(x+1) on interval [0,2] and n=100 using the calculator is 0.90139_.

### Explanation of Solution

The value of the function f(x)=x(x+1) is a increasing function. The left Riemann Sum gives the lower estimate and right Riemann Sum gives upper estimate Thus, get the expression as follows:

0.8946<02xx+1dx<0.9081

Given:

The function as f(x)=x(x+1)

The region lies between x=0 and x=2. So the limits are a=0 and b=2.

Number of rectangles n=100.

Calculation:

Show the Equation of the function f(x) as follows:

f(x)=x(x+1) (1)

Calculate the value of the function f(x) for different values of x within the interval [0,2].

Substitute 0 for x in Equation (1).

f(0)=0(0+1)=0

Substitute 1.5 for x in Equation (1).

f(1.5)=1.5(1.5+1)=0.6

Substitute 2 for x in Equation (1).

f(2)=2(2+1)=0.666

The calculated values of x shows that the function f(x)=x(x+1) is aincreasing function within the interval [0,2].

The expression to find the left Riemann sum Ln as shown below:

Ln=i=1nf(xi)Δx=f(x0)Δx+f(x1)Δx+...+f(xn1)Δx (2)

Here, the left endpoint height of first rectangle is f(x0), the width is Δx, height of left endpoint of second rectangle is f(x1), and left endpoint height of nth rectangle is f(xn1).

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.

Substitute 2 for b, 0 for a and 100 for n in Equation (3).

Δx=20100=0.02

The value of left and right end points within the interval [0,2] are shown below:

The left endpoints are x0=0, x1=0.02, x2=0.04x98=1.96 and x99=1.98.

The right endpoints are x1=0.02, x2=0.04, x3=0.06x99=1.98 and x100=2.

Calculate the left Riemann sum using calculator.

Substitute 0.02 for Δx, 100 for n, and values of left end points in Equation (2).

L100=i=1100f(xi)Δx=f(x0)Δx+f(x1)Δx+...+f(x99)Δx=f(0)×0.02+f(0.02)×0.02+...+f(1.98)×0.02=0.89469

The value of left Riemann Sum is 0.89469_.

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 right Riemann sum using calculator.

Substitute 0.02 for Δx, 100 for n, and values of right end points in Equation (4).

Rn=i=1100f(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(x100)Δx=f(0.02)×0.02+f(0.04)×0.02+...+f(2)×0.02=0.90802

The value of right Riemann Sum is 0.90802_.

Calculate the value of integral 02xx+1dx using a calculator.

02xx+1dx=0.90139 (5)

Compare the left and right Riemann Sum value with value of integral 02xx+1dx.

Since, the function f(x)=1(x+1) is a increasing function.

The value of left Riemann sum gives a lower estimate.

The value of the right Riemann sum gives a upper estimate.

Thus, expression the left and right Riemann sum as follows:

0.89469<02xx+1dx<0.90802

Thus, the expression for left and right Riemann sum is shown as 0.89469<02xx+1dx<0.90802.

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