Improper integral and rectangle method

[lokofer]

Improper integral and "rectangle" method

If we have a definite integral then..using "rectangle" method we can get the approximation:

[tex] \int_{a}^{b}f(x)dx \sim \sum_{n=0}^{N}f(a+nh)h [/tex]

My question is..how do you define this method when b-->oo (Imporper integral?)...

 

[AKG]

The upper limit N must be such that a + Nh = b. If b -> oo, then make the upper limit N = oo.

 

[tacman]

The "rectangle" method is a numerical method for approximating the value of a definite integral by dividing the interval of integration into smaller subintervals and using the height of rectangles to estimate the area under the curve. This method is useful for integrals that cannot be evaluated analytically or for which the antiderivative is difficult to obtain.

However, for improper integrals, where one or both of the limits of integration are infinite, this method cannot be directly applied. In this case, we need to consider a limiting process where the upper limit of integration, b, approaches infinity. This is known as an improper integral.

To approximate an improper integral using the "rectangle" method, we can choose a large upper limit of integration, say b = N, and then use the "rectangle" method to evaluate the integral from a to N. Then, as N approaches infinity, our approximation will become more accurate.

It is important to note that this is only an approximation and the actual value of the improper integral may be different. Therefore, it is necessary to check the convergence of the integral by evaluating its limit as the upper limit of integration approaches infinity.

In summary, the "rectangle" method can be extended to approximate improper integrals by considering a limiting process where the upper limit of integration approaches infinity. This allows us to approximate the value of the integral, but we must also check for convergence to ensure the accuracy of our result.