Comparison test for improper integrals - integration, Mathematics

Assignment Help:

Comparison Test for Improper Integrals

Here now that we've seen how to actually calculate improper integrals we should to address one more topic about them.  Frequently we aren't concerned along with the actual value of these integrals.  In place of it we might just only be interested in if the integral is convergent or divergent.  As well, there will be some integrals which we simply won't be capable to integrate and yet we would still such as to know if they converge or diverge.  

 To deal along with this we have got a test for convergence or divergence which we can use to assist us answer the question of convergence for a not proper integral. 

We will provide this test only for a sub-case of the infinite interval integral, though versions of the test exist for the other sub-cases of the infinite interval integrals also integrals with discontinuous integrands.

Comparison Test

If f (x) ≥ g (x) > 0 on the interval [a, ∞] then,

1. If ∫a f(x) converges then so does ∫a g(x) dx.

2. If ∫a g(x) dx diverges then so does ∫a f (x) dx.

Note: If you think in terms of area the Comparison Test makes a lot of sense. Determine if f (x) is larger than g(x) then the area within f (x) must as well be larger than the area under g(x). Thus, if the area within the larger function is finite after that the area under the smaller function has to be finite. Similarly, if the area under the smaller function is infinite after that the area within the larger function must as well be infinite. Be cautious not to misuse this test. If the smaller function converges there is no basis to believe that the larger will as well converge (after all infinity is larger as compared to a finite number...) and determine if the larger function diverges there is no reason to believe that the smaller function will also diverge.


Related Discussions:- Comparison test for improper integrals - integration

Mensuration of plane figures, a sail has a spread of canvas as measured 12'...

a sail has a spread of canvas as measured 12'',12'', 15'' and 9'' and it has 90 degrees. Find the area of one side of the sail

Calenders, on which date of the week does 4th december 2001 falls?

on which date of the week does 4th december 2001 falls?

Geometry help, One of two complementary angles is 80& of the other. What is...

One of two complementary angles is 80& of the other. What is the degree measure of the smaller angle?

Find the Regular Grammar for the following Regular Expressio, Find the Regu...

Find the Regular Grammar for the following Regular Expression: a(a+b)*(ab*+ba*)b.

6, 200000+500

200000+500

Algebra, let setM={X,2X,4X} for any numberX .if average (arthemetic mean)of...

let setM={X,2X,4X} for any numberX .if average (arthemetic mean)of the number in setM is 14.what is the value of X?

Relations, Suppose A and B be two non-empty sets then every subset of A Χ B...

Suppose A and B be two non-empty sets then every subset of A Χ B describes a relation from A to B and each relation from A to B is subset of AΧB. Normal 0 fals

Find out the hydrostatic force on the triangular plate, Find out the hydros...

Find out the hydrostatic force on the following triangular plate that is submerged in water as displayed. Solution The first thing to do here is set up an axis system

Introduction to learning to count, INTRODUCTION : Most of us, when plannin...

INTRODUCTION : Most of us, when planning the first mathematical experience for three-year olds, think in terms of helping them memorise numbers from 1 to 20. We also teach them to

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd