Approximation and Simpson's Rule

In summary: Trapezoidal_ruleIn summary, the problem asks for estimates of a definite integral using the left, trapezoid, and Simpson's rules with given values for the left rule at 40 and the trapezoid rule at 40. Using the concept of approximate errors, the error for the trapezoid rule at 40 is found and used to estimate the error for an n-value of 120. There are specific formulas for each rule, which can be found in a textbook or online.
  • #1
ada0713
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Homework Statement



Suppose the exact value of a particular definite integral is 6. The following questions refer to estimates of this integral using the left, trapezoid, and Simpson's rules. Use what you know about approximate errors to answer the following questions. Give your answer to 4 decimal places.

(a) Suppose Left(40)=2.3055. Estimate Left(120)


(b) Suppose Trap(40)=2.7250. Estimate Trap(120)


(c) Suppose Simp(40)=3.1680. Estimate Simp(120)



The Attempt at a Solution



I know that Actual- Approximated= Error, but I am not sure how to use this concept
to solve the problem. I've tried to find the ratio of each, but not enough information
was given in order to estimate what it asked for. (In terms of "not enough information",
I mean--- Shouldn't they give us the actual function whose definite integral is 6?)
Please help me with the start and I'll try to figure out he rest!
 
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  • #2
Anyone know how to do this problem? HELP!
 
  • #3
Error in trap(40) = 6 - 2.7250 = K2(b-a)^3/(12)(40)^2. Therefore, K2(b-a)^3/12 = 40^2(6-2.7250).
Then trap(120) = 40^2(6-2.7250)*(1/(120)^2
 
  • #4
What does K2 stand for? I don't quite understand how you did it..
 
  • #5
[tex]\left|E_{T}\right| = \frac{K_{2}(b-a)^{3}}{12n^{2}}[/tex]

Where [tex]K_{2} \geq \left|f''(x)\right| [/tex] on [a,b]

Since we know the exact value of the integral, we can subtract the approximation given to get a value for the error of T(40). In this case 6 - 2.7250 = 3.275.

3.275 =[tex] \frac{K_{2}(b-a)^{3}}{12(40)^{2}}[/tex]

Multiply both sides by 40^2.

(40^2)(3.275) = [tex] \frac{K_{2}(b-a)^{3}}{12}[/tex]

Now, use this fact to find the error when n = 120.
 
  • #6
Oh.. Thanks..
But it seems the formula up there is only useful when we find the approximation for
trapezoide. Are there specific formulas for left rule and simpsons's rule as well?
This is new to me so I tried to find the concept in the textbook but I don't think it mentions this :(
 
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  • #7

FAQ: Approximation and Simpson's Rule

1. What is approximation and why is it important in science?

Approximation is the process of estimating a value or solution based on incomplete or imperfect information. It is important in science because many real-world problems cannot be solved exactly, so approximation allows us to make reasonable and useful estimates.

2. What is Simpson's Rule and how is it used in scientific calculations?

Simpson's Rule is a numerical method for approximating the area under a curve. It is used in scientific calculations when the exact solution cannot be obtained or is too complex to compute. It provides a more accurate estimate than simpler methods like the trapezoidal rule.

3. What are the limitations of Simpson's Rule?

Simpson's Rule is limited by the fact that it can only be applied to functions that are continuous and have a smooth curve. It is also limited by the number of intervals used, as a larger number of intervals will provide a more accurate estimate. Additionally, it may not be applicable to certain types of non-linear functions.

4. How is Simpson's Rule related to the concept of integration?

Simpson's Rule is a numerical method for approximating the area under a curve, which is the same concept as integration. Integration is the mathematical process of finding the area under a curve, while Simpson's Rule is a technique for approximating this area.

5. Can Simpson's Rule be used for any type of curve?

No, Simpson's Rule can only be used for curves that are continuous and have a smooth shape. It may not be applicable to curves with abrupt changes or sharp corners. In these cases, other numerical methods or analytical techniques may be more suitable for finding the area under the curve.

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