 # Chapter 7 – Techniques of Integration

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Chapter 7 – Techniques of Integration
7.7 Approximation Integration 7.7 Approximation Integration Erickson

Why do we use Approximate Integration?
There are two situations in which it is impossible to find the exact value of a definite integral: When finding the antiderivative of a function is difficult or impossible If the function is determined from a scientific experiment through instrument readings or collected data. There may not be a formula for the function. 7.7 Approximation Integration Erickson

Approximate Integration
For those cases we will use Approximate Integration We have used approximate integration on chapter 5 when we learned how to find areas under the curve using the Riemann Sums. We used Left, Right and Midpoint rules. Now we are going to learn two new methods: The Trapezoid Rule and Simpson’s Rule Let’s compare the approximation methods. 7.7 Approximation Integration Erickson

Midpoint Rule Remember, the midpoint rule states that where and
7.7 Approximation Integration Erickson

Trapezoidal Rule The Trapezoid rule approximates the integral by averaging the approximations obtained by using the Left and Right Endpoint Rules: where and 7.7 Approximation Integration Erickson

Example 1 Use (a) the Midpoint Rule and (b) the Trapezoidal Rule with n = 5 to approximate the integral below. Round your answer to six decimal places. 7.7 Approximation Integration Erickson

Error Bounds in MP and Trap Rules
Suppose for a ≤ x ≤ b. If ET and EM are the errors in the Trapezoidal and Midpoint Rules, then 7.7 Approximation Integration Erickson

Example 2 Find the error in the previous problem. Previous problem:
Use (a) the Midpoint Rule and (b) the Trapezoidal Rule with n = 5 to approximate the integral 7.7 Approximation Integration Erickson

Example 3 How large should we take n in order to guarantee that the Trapezoidal Rule and Midpoint Rule approximations are accurate to within for the integral below? 7.7 Approximation Integration Erickson

Simpson’s Rule Simpson’s Rule uses parabolas to approximate integration instead of straight line segments. where and n is even. 7.7 Approximation Integration Erickson

Error Bounds in Simpson’s Rule
Suppose for a ≤ x ≤ b. If ES is the error involved using Simpson’s Rule, then 7.7 Approximation Integration Erickson

Example 4 Use the (a) Midpoint Rule and (b) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. Compare your results to the actual value to determine the error in each approximation. 7.7 Approximation Integration Erickson

Example 5 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and
(c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. 7.7 Approximation Integration Erickson

Example 6 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and
(c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. 7.7 Approximation Integration Erickson

Example 7 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and
(c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. 7.7 Approximation Integration Erickson

Example 8 (a) Find the approximations T10 and M10 for the above integral. (b) Estimate the errors in approximation of part (a). (c) How large do we have to choose n so that the approximations Tn and Mn to the integral part (a) are accurate to within ? 7.7 Approximation Integration Erickson

Example 9 The table (supplied by San Diego Gas and Electric) gives the power consumption P in megawatts in San Diego County from midnight to 6:00 AM on December 8, Use Simpson’s Rule to estimate the energy used during that time period. (Use the fact that power is the derivative of energy.) t P 0:00 1814 3:30 1611 0:30 1735 4:00 1621 1:00 1686 4:30 1666 1:30 1646 5:00 1745 2:00 1637 5:30 1886 2:30 1609 6:00 2052 3:00 1604 7.7 Approximation Integration Erickson

Book Resources Video Examples More Videos Example 2 – pg. 510
Using the Midpoint Rule to approximate definite integrals, Part I Using the Midpoint Rule to approximate definite integrals, Part 2 Using the Midpoint Rule to approximate definite integrals, Part 3 The Trapezoidal Rule Using the Trapezoidal Rule to approximate an integral Errors in the Trapezoidal Rule and Simpson’s Rule Simpson’s Rule Using Simpson’s Rule to Approximate an Integral 7.7 Approximation Integration Erickson

Book Resources Wolfram Demonstrations
Comparing Basic Numerical Integration Methods 7.7 Approximation Integration Erickson