Integral Approximation/Error Analysis

In summary, the conversation is about a problem involving approximating an integral using either Riemann Sum or Trapezoidal Sum with a specific accuracy requirement. The difficulty lies in finding the appropriate value for n, as the function has a steep slope and the required mesh near the endpoint is very fine. It is noted that the trapezoidal rule tends to overestimate the value due to the concave up nature of the function. The speaker was surprised to find that a relatively low value of n (61,000) worked for their approximation.
  • #1
rwsjyiy
5
0
Hi so i have this problem that I am working on but I am having trouble with it.
I have to approximate the integral using either Riemann Sum or Trapezoidal sum with .2 of the actual integral. It is the integral of e^(x^2+3x+1) from 0 to 3.
Every time i try to find the n value, it ends up being huge! so it gives me a really small delta x number. Any Help?
Thanks
 
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  • #2
It is not surprising. The exponent is 19 at the end point, so the function is very large and has a steep slope. You need a very fine mesh near x=3. Is the .2 absolute or relative? If absolute, your result (n huge!) is not surprising.
 
  • #3
it says approximate the integral within.2 and be at least as great as the actual integral
 
  • #4
f(x) is concave up so the trapezoidal rule always overestimates the value. I was actually surprised that I only got n = 61,000 works lol, thought it would be higher.
 
Last edited:

1. What is integral approximation?

Integral approximation is a method used in mathematics and science to estimate the value of an integral (the area under a curve) when it is not possible or practical to find the exact value. It involves breaking down the integral into smaller, simpler components and using mathematical techniques to approximate their values.

2. Why is integral approximation important?

Integral approximation is important because it allows us to estimate the value of integrals that cannot be solved analytically. This is useful in many fields of science and engineering, where integrals are often used to model real-world phenomena.

3. What are the different methods of integral approximation?

There are several methods of integral approximation, including the trapezoidal rule, Simpson's rule, and the midpoint rule. These methods differ in their level of accuracy and complexity, but all involve breaking the integral into smaller parts and using numerical techniques to estimate their values.

4. How do you choose the best method for integral approximation?

The best method for integral approximation depends on the specific integral being approximated and the desired level of accuracy. In general, more complex methods such as Simpson's rule will provide a more accurate approximation than simpler methods like the midpoint rule. It is also important to consider the computational resources available, as some methods may require more time and computing power.

5. What is error analysis in integral approximation?

Error analysis in integral approximation involves evaluating the difference between the approximated value and the exact value of the integral. This allows us to determine the accuracy of our approximation and make adjustments as needed. Error analysis is an important step in ensuring the validity of integral approximations.

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