Methods for Estimating Integrals without Calculators

  • Thread starter intenzxboi
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In summary, the author suggests that you can approximate an integral by hand by using Simpson's rule, which involves calculating the values of the function at certain points.
  • #1
intenzxboi
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Homework Statement



estimate int 1/ln x between 2 and 4


well i thought that for this to work you would have to integrate first and then find f(4) - f(2)

but how come right now we are learning about using the midpoint, left, right, trap and simpson rule.

but instead of integrating it first they are just using the equation without integrating first??

also for int 1/ln x between 2 and 4 using one of the estimating methods(ex. right) is there any way to do this problem without a calculator??
 
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  • #2
Well, good luck on finding an antiderivative for 1/(ln x)!

Otherwise, you're going to be stuck with calculating an estimate for the integral, rather than an exact value.

You'll need a calculator to do your estimate.
 
  • #3
You can approximate it by hand.

1. Find the the first, second, and possibly third derivatives.
2. Evaluate at x = 2. Call them a = f ''(2), b = f '(2), c = f (2)
3. Then call 1/ln(x) = c + b(x-2) + a(x-2)^2.

Then integrate the result from step 3, and that will be a good approximation.
 
  • #4
csprof2000 said:
You can approximate it by hand.

1. Find the the first, second, and possibly third derivatives.
2. Evaluate at x = 2. Call them a = f ''(2), b = f '(2), c = f (2)
3. Then call 1/ln(x) = c + b(x-2) + a(x-2)^2.

Then integrate the result from step 3, and that will be a good approximation.

Sure, you don't need a calculator for that estimate, but if the OP doesn't know about Taylor series, this will be pure magic.
 
  • #5
intenzxboi said:

Homework Statement



estimate int 1/ln x between 2 and 4well i thought that for this to work you would have to integrate first and then find f(4) - f(2)

but how come right now we are learning about using the midpoint, left, right, trap and simpson rule.

but instead of integrating it first they are just using the equation without integrating first??

also for int 1/ln x between 2 and 4 using one of the estimating methods(ex. right) is there any way to do this problem without a calculator??

Since you mentioned http://en.wikipedia.org/wiki/Simpson_rule" [Broken] this may be the way you are supposed to do it. Note that the formula does not involve taking the antiderivative of 1/ln x, you only need the values of that function at 2,3,4.
 
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  • #6
ic ic thanks
 

1. How do you estimate the accuracy of your estimate?

To estimate the accuracy of an estimate, you can use statistical methods such as confidence intervals or margin of error. These calculations take into account the variability in the data and provide a range in which the true value is likely to fall within.

2. What factors should be considered when making an estimate?

When making an estimate, it is important to consider factors such as the quality and quantity of data available, the assumptions made in the estimation process, and the expertise of the person making the estimate. Other external factors such as economic conditions or market trends may also play a role.

3. How do you handle uncertainty in your estimates?

Uncertainty is inevitable in any estimate, and it is important to acknowledge and account for it. One way to handle uncertainty is to use a range of values rather than a single point estimate. Another approach is to use sensitivity analysis, which evaluates how changes in different variables can affect the final estimate.

4. Can estimates be biased?

Yes, estimates can be biased if the data used to make the estimate is not representative or if the person making the estimate has a biased perspective. It is important to be aware of potential biases and try to minimize their impact on the estimate by using unbiased data and involving multiple perspectives in the estimation process.

5. How often should estimates be updated?

The frequency of updating estimates depends on the specific situation and the purpose of the estimate. In rapidly changing environments, estimates may need to be updated more frequently to remain accurate. However, in more stable situations, less frequent updates may be sufficient. It is important to regularly review and reassess estimates to ensure their accuracy and relevance.

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