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

Find the equation of circle concentric – coordinate geometry, 1. A point P(...

1. A point P(a,b) becomes (3,c) after reflection in x - axis, and (d,6) after reflection in the origin. Show that a = 3, b = - 6, c = 6, d = 2 2. If the pair of lines ax² + 2pxy

Explain the vertex formula, Explain the Vertex Formula ? The vertex for...

Explain the Vertex Formula ? The vertex formula is a convenient way of finding the vertex of the graph for any quadratic function. The graph of the quadratic equation f(x) = ax

Algebraic expression, i dont understand what my teacher disccussing thats w...

i dont understand what my teacher disccussing thats why i want to learn for this lesson. i want to ask'' what is the variables?

Project, report on shares and dovidend using newspaer

report on shares and dovidend using newspaer

What is pythagorean triples, What is Pythagorean Triples? A set of thre...

What is Pythagorean Triples? A set of three numbers a, b, and c that can satisfy the equation A 2 +b 2 = c 2 , is called a Pythagorean triple. The following is a list of

Compute the quartile coefficient of skewness, By using the above data compu...

By using the above data compute the quartile coefficient of skewness Quartile coefficient of skewness = (Q3 + Q1 - 2Q2)/(Q3 + Q1)                                The positio

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