convergence of the integral example 1

Register or Login to View the Solution or Ask a Question

Introduction: In this solution, we have explained what is convergence of the integral example 1. We will test the convergence of the integral 0 to a 1/(sqrt(a-x)dx.

Qestion : Examine the convergence of the integral

\[\int_{0}^{a}\frac{dx}{\sqrt{a-x}}\]

Solution :

Let

\[I=\int_{0}^{a}\frac{dx}{\sqrt{a-x}}\]

Since integrand here is

\[\frac{1}{\sqrt{a-x}}\]

which is unbounded on Interval of integration [0, a] . So I is an improper integral of second kind.

\[I=limit_{\epsilon\rightarrow0+}\int_{0}^{a-\epsilon}\frac{dx}{\sqrt{a-x}}\]

\[=limit_{\epsilon\rightarrow0+}\int_{0}^{a-\epsilon}{\left(a-x\right)^{-\frac{1}{2}}dx}\]

\[=limit_{\epsilon\rightarrow0+}\left[-\frac{\left(a-x\right)^\frac{1}{2}}{\frac{1}{2}}\right]_0^{a-\epsilon}\]

\[=-2limit_{\epsilon\rightarrow0+}\left[\epsilon^\frac{1}{2}-a^\frac{1}{2}\right]\]

\[=-2\left(0^\frac{1}{2}-a^\frac{1}{2}\right)\]

\[=2a^\frac{1}{2}\] \[\int_{0}^{a}\frac{dx}{\sqrt{a-x}}\]

\[=2\sqrt{a}\]

thus we get

\[\int_{0}^{a}\frac{dx}{\sqrt{a-x}}=2\sqrt{a}\]

which is a finite number that is why given integral will be convergent.


Register or Login to View the Solution or Ask a Question

Comments

No comments yet. Why don’t you start the discussion?

Leave a Reply