Math Is Fun Forum

IEd

Guest

How can i modify the fundamental error?

So, there is this calculus problem (first seen in bluepenredpen) where there is a definite integral :-

∫1/x^2 under limits ( -2 ---> 1)

its quite simple as the answer is -3/2 but here's the thing, when put into an integral calculator , it says that the given function is divergent. Also , the function is not continuous as it passses x=0 which gives -1/0 which is not defined.

HOW CAN I SOLVE such problems, is there any modification to counter such error. it could be possible to say that its an incomplete question as some problems would have limitations written (where x≠0).

Bob

Administrator

Registered: 2010-06-20

Posts: 10,163

Re: How can i modify the fundamental error?

hi IEd

Welcome to the forum.

You can use the usual rules to integrate the function and substitute values but that doesn't make your answer correct.  For any integral you should always make a sketch of the function so you can check for places where the function changes sign and where the function goes to infinity.

This function is 'infinite' at x = 0 and the area under the graph cannot be determined.  I suppose it might tend to a limit but try here:
https://www.wolframalpha.com/input/?i=i … -2+to+%2B1 and you'll see that doesn't happen here.

So I don't think any tricks exist to help.

Even if you said x not zero you'd still encounter a problem if the area tends to infinity.

One way to examine this would be to create a program or use a spreadsheet to calculate the area of rectangles that are approximately vertical strips from the graph.  What happens to the size of these as you approach x = 0.  (Probably enough to just do right hand side values).

Here's my attempt:

I created row headings and then entered x = 0.1 into cell A2
I made B2 = A2/10 so that I am dividing the region between that x value and zero into 10 strips.

I made A3 = A2 minus $B$2.  The dollar signs mean that B2 will always be used rather than advancing the reference when I copy it

So I could then copy the formula down the column.

Then I calculated the y values and the area of each strip. 

So it was easy to reduce the A2 value towards zero and see what happens.  Here's a screen shot when x = 0.001 with the formula view alongside so you can see how each line is calculated.

This is using MS Excel.  If you have that software you can try this yourself. As x gets smaller the area values tend to infinity as expected.

This is because, for this function, 1/x^2 gets bigger more rapidly than dx gets smaller.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob