   Chapter 5.6, Problem 33E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Numerical Approximation Use the Midpoint Rule with n = 4 to approximate π, where π = ∫ 0 1 4 1 + x 2   d x . π = ∫ 0 1 4 1 + x 2   d x . Then use a graphing utility to evaluate the definite integral. Compare your results.

To determine

To prove: The value of the integral π=0141+x2dx using the midpoint rule with n=4 and a graphing utility to approximate π and then compare the result.

Explanation

Given Information:

The provided integral is π=0141+x2dx.

Proof:

Consider the integral, π=0141+x2dx

Here, the interval is [0,1], which is divided into n=4 subintervals.

Evaluate the width of each interval by dividing the difference of interval by n.

Δx=104=14

Therefore, the subintervals are [0,14],[14,24],[24,34],[34,1]

Evaluate the midpoint of the subinterval [0,14],

[0,14]=0+142=0.125

Evaluate the midpoint of the subinterval [14,24],

[14,24]=14+242=342=0.375

Evaluate the midpoint of the interval [24,34],

[24,34]=24+342=542=0.625

Evaluate the midpoint of the interval [34,1],

[34,1]=34+12=742=0

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