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Calculus II - Section 7.7 #24: Approximate Integration

Introduction

Approximate integration techniques are essential tools when we encounter definite integrals that cannot be evaluated exactly using the Fundamental Theorem of Calculus. Section 7.7 introduces several methods for numerical integration, including the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule.

Problem #24

Problem Statement

Use the following methods to approximate the integral e dx with an error less than or equal to 0.001:

Trapezoidal Rule
Midpoint Rule
Simpson's Rule

Solution Approach

Let's examine the integral we're approximating:

I = e dx

This integral cannot be expressed in terms of elementary functions, making numerical approximation necessary. To achieve the required precision ( 0.001), we must determine the appropriate number of subintervals (n) for each method using their respective error estimation formulas:

Applying the Trapezoidal Rule

Finding the Required Number of Subintervals

For f(x) = e, we first find f''(x):

f'(x) = 2xe

f''(x) = e(4x + 2)

On [0,2], the maximum value of |f''(x)| occurs at x=2:

K = f''(2) = 18e 982.656

Using the error bound formula for the Trapezoidal Rule:

|E| K(b-a)/12n

Solving for n with |E| 0.001:

982.656(2-0)/12n 0.001

n 982.6568/120.001 655,104

n 655,104 809.4

Thus, we need at least n = 810 subintervals for the Trapezoidal Rule.

Implementing the Trapezoidal Rule

The Trapezoidal Rule formula is:

f(x)dx (b-a)/2n [f(x) + 2f(x) + 2f(x) + ... + 2f(x) + f(x)]

With n = 810 and x = (2-0)/810, applying the formula yields approximately:

e dx 16.452

Applying the Midpoint Rule

The error bound for the Midpoint Rule requires fewer subintervals:

|E| K(b-a)/24n

Solving for n:

982.656(2-0)/24n 0.001

n 327,552 572.3

Thus, we need at least n = 573 subintervals for the Midpoint Rule.

The Midpoint Rule formula is:

f(x)dx (b-a)/n [f(m) + f(m) + ... + f(m)]

where mi is the midpoint of the ith subinterval.

With n = 573, applying the formula yields approximately:

e dx 16.453

Applying Simpson's Rule

Finding the Fourth Derivative

For Simpson's Rule, we need f(x):

f(x) = e(16x + 48x + 12)

On [0,2], the maximum value of |f(x)| occurs at x=2:

K = f(2) = 460e 25,101.4

Determining the Required Subintervals

Using the error bound formula for Simpson's Rule:

|E| K(b-a)/180n

Solving for n:

25,101.4(2-0)/180n 0.001

n 7,136.4 9.2

Since Simpson's Rule requires an even number of subintervals, we need n = 10.

Implementing Simpson's Rule

Simpson's Rule formula is:

f(x)dx (b-a)/3n [f(x) + 4f(x) + 2f(x) + 4f(x) + ... + 4f(x) + f(x)]

Notice the pattern of coefficients: 1,4,2,4,2,...,4,1

With n = 10, we calculate the following values:

i	xi	f(xi)	Weight	Weighted Value
0	0	1	1	1
1	0.2	1.0408	4	4.1632
2	0.4	1.1735	2	2.3470
3	0.6	1.4333	4	5.7332
4	0.8	1.8965	2	3.7930
5	1.0	2.7183	4	10.8732
6	1.2	4.2207	2	8.4414
7	1.4	7.0993	4	28.3972
8	1.6	12.9358	2	25.8716
9	1.8	25.5335	4	102.1340
10	2.0	54.5982	1	54.5982

Sum of weighted values: 246.3518

Applying Simpson's Rule:

e dx (2-0)/310 246.3518 16.453

Comparison and Analysis

The results from our three methods are:

Trapezoidal Rule (n=810): 16.452
Midpoint Rule (n=573): 16.453
Simpson's Rule (n=10): 16.453

Key Observation: Simpson's Rule achieved the requested precision with only 10 subintervals, while the Trapezoidal Rule required 810 and the Midpoint Rule required 573. This demonstrates the superior efficiency of Simpson's Rule for smooth functions like e.

Conclusion

Problem #24 illustrates the practical application of approximate integration techniques. Each method has advantages:

Simpson's Rule: Most efficient for smooth functions
Midpoint Rule: Generally requires about half the subintervals of the Trapezoidal Rule for the same accuracy
Trapezoidal Rule: Simple to implement but often requires many more subintervals

Common Mistakes to Avoid

Forgetting to check the maximum value of derivatives over the entire interval when calculating error bounds
Not using an even number of subintervals for Simpson's Rule
Calculation errors when implementing the formulas, especially with alternating coefficients in Simpson's Rule
Not recognizing when a function's behavior requires special consideration (e.g., sharp changes in slope)