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Integral Calculus

Cool goIITian

Joined:	8 Jul 2008
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17 Mar 2010 15:14:44 IST

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Integrate

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Integrate from 1 to 2.5 [ x ] dx where [ ] is greatest integer function.

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mardav, NIT-B civil engineering.

Blazing goIITian

Joined: 9 Feb 2009

Posts: 619

17 Mar 2010 15:31:55 IST

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integrate it by dividing the limit into intervals. (0,1) then (1,2) and last (2,2.5). ur answer will be 0+1+1 i.e 2

HIMANSHU JAIN

Scorching goIITian

Joined: 8 Feb 2007

Posts: 267

17 Mar 2010 16:59:13 IST

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Try to analyze it's graph.

Check the graph on this link:

http://library.thinkquest.org/20991/alg2/graphs.html

now, u'll get to know that [x] is 1 for (1,2) and 2 for (2,2.5).

so, u'll get :

1.dx for (1,2) = 1

and 2.dx for (2,2.5) =1.

So, ans is 2.

Do ask again if any doubt is there.

Deepak Aggarwal

Blazing goIITian

Joined: 9 May 2009

Posts: 766

18 Mar 2010 11:17:52 IST

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This is the graph of greatest integer function (or the floor function).

Now break the limit for the integral first from 1 to 2 and then from 2 to 2.5.

I = int x from 1 to 2.5 [x] . dx

I = int x from 1 to 2 [x] . dx + int x from 2 to 2.5 [x] . dx

I = int x from 1 to 2 (1) . dx + int x from 2 to 2.5 (2) . dx

I = limit from 1 to 2 (x) + limit from 2 to 2.5 (2x)

I = (2 - 1) + 2(2.5 - 2) = 2 (Ans)

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