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 July 6th, 2017, 10:36 AM #1 Member   Joined: Apr 2017 From: PA Posts: 45 Thanks: 0 Find the definite integral Find the definite integral $$\int_{0}^{20}|X-[X+.5]|\,dx,$$ (where X is the greatest integer for [X+.5]) Attempt: I found out that if X> or = to X+.5, then we would have X-[X+.5] as our function. And if X
July 6th, 2017, 11:48 AM   #2
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Quote:
 Originally Posted by poopeyey2 (where X is the greatest integer for [X+.5])
what does this mean?

do you mean that

$[X+.5]$ is The Floor of $X+.5$, usually written $\lfloor X+.5 \rfloor$ ?

July 6th, 2017, 11:55 AM   #3
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Quote:
 Originally Posted by romsek what does this mean? do you mean that $[X+.5]$ is The Floor of $X+.5$, usually written $\lfloor X+.5 \rfloor$ ?
If so if you plot $|X - \lfloor X+.5 \rfloor|$

you'll see it's 20 isosceles triangles each $0.5$ tall, and $1$ wide

The area of each triangle is just $\dfrac 1 2 (1)(0.5) = \dfrac 1 4$

So 20 of these sum to an area of $5$

 July 6th, 2017, 03:34 PM #4 Member   Joined: Apr 2017 From: PA Posts: 45 Thanks: 0 Yes, you wrote it correctly. Last edited by skipjack; July 7th, 2017 at 02:43 AM.
 July 7th, 2017, 03:12 AM #5 Global Moderator   Joined: Dec 2006 Posts: 18,166 Thanks: 1424 Your attempt (if X $\small\geqslant$ X + .5, etc.) was obviously incorrect.

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