1. Chapter 7 – Techniques of Integration
7.7 Approximation Integration
7.7 Approximation Integration
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2. 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
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3. 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
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4. Midpoint Rule Remember, the midpoint rule states that where and
7.7 Approximation Integration
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5. 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
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6. 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
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7. 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
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8. 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
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9. 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
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10. 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
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11. 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
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12. 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
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13. 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
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14. 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
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15. 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
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16. 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
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17. 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
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18. 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
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19. Book Resources Wolfram Demonstrations
Comparing Basic Numerical Integration Methods
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20. Web Links http://youtu.be/JGeCLfLaKMw http://youtu.be/z_AdoS-ab2w
7.7 Approximation Integration
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