Need help with numerical integration

In summary, numerical integration is a mathematical technique used to approximate the value of an integral by dividing it into smaller sections and calculating the sum of those sections. It is important because it allows us to estimate integrals when an exact solution is not possible. The most commonly used methods are the Trapezoidal rule, Simpson's rule, and the Midpoint rule. The accuracy of numerical integration depends on the method and number of divisions chosen. To minimize errors, it is important to choose an appropriate method and use a sufficient number of intervals, along with high precision arithmetic and checking results using different methods or increasing the number of intervals.
  • #1
undefined83
8
0
Im supposed to solve
integral 10 to +infinity ((sin(1/x)/(1+x^3))dx with error precision of e=0.5*10^-4. Can someone please give me detailed explenation of solving this. (Supposedly by Simpson but i get lost in the way.

P.S. sorry for bad spelling and lack of proper formula notions.
 
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  • #2
I process from integral i get
1/3[ln((x+1)/sqrt(x^2-x+1))+sqrt(3)*arctg(2*sqrt(3)*x/3-sqrt(3)/3)] from M to +infinity <=1/4*10^-2

than i get

-ln((M+1)/(sqrt(M^2-M+1))) +sqrt(3)*(pi/2-arctg(2*sqrt(3)*M/3-sqrt(3)/3))<=7.5*10^-3

and i can't find any exact solution to solve that
 
  • #3


Numerical integration is a method used to approximate the value of an integral when it cannot be solved analytically. In this case, we are trying to solve the integral from 10 to positive infinity of (sin(1/x)/(1+x^3))dx with an error precision of e=0.5*10^-4. This means that we want our approximation to be within 0.5*10^-4 of the actual value of the integral.

To solve this using Simpson's rule, we need to first divide the interval of integration (10 to infinity) into smaller subintervals. Simpson's rule works by approximating the curve of the function with a series of parabolas, so we need to have an even number of subintervals. Let's say we divide the interval into n subintervals, each with a width of h.

Next, we need to calculate the values of the function at the endpoints of each subinterval. In this case, our function is (sin(1/x)/(1+x^3)), so we need to calculate f(10), f(10+h), f(10+2h), and so on until we reach the endpoint of the last subinterval. We can use a calculator or a computer program to do this.

Once we have all the function values, we can use Simpson's rule formula to approximate the value of the integral. The formula is:

I ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(xn-1) + f(xn)]

where x0 is the starting point of the interval (10 in this case), xn is the endpoint of the interval (infinity in this case), and h is the width of each subinterval (h = (infinity - 10)/n).

To meet our error precision of e=0.5*10^-4, we need to choose a value of n that will make the error term (h^4/180)*f^(4)(ξ) (where ξ is some value between 10 and infinity) less than or equal to 0.5*10^-4. This will require some trial and error, but we can start by choosing a value of n and calculating the error term. If it is larger than 0.5*
 

1. What is numerical integration?

Numerical integration is a mathematical technique used to estimate the value of an integral (area under a curve) by dividing it into smaller sections and calculating the sum of the areas of those sections.

2. Why is numerical integration important?

Numerical integration is important because it allows us to approximate the value of an integral when an exact solution is not possible or difficult to obtain. It is also used in many scientific and engineering applications where analytical solutions are not available.

3. What are the different methods of numerical integration?

The most commonly used methods of numerical integration are the Trapezoidal rule, Simpson's rule, and the Midpoint rule. These methods differ in the way they divide the integral into smaller sections and calculate the sum of those sections.

4. How accurate is numerical integration?

The accuracy of numerical integration depends on the method used and the number of divisions or intervals chosen. Generally, the more divisions used, the more accurate the approximation will be. However, using too many divisions can also lead to errors due to rounding and truncation.

5. How can errors in numerical integration be minimized?

To minimize errors in numerical integration, it is important to choose an appropriate method for the given integral and to use a sufficient number of intervals. It is also helpful to use a computer or calculator with high precision arithmetic and to check the results using different methods or increasing the number of intervals.

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