July 6th, 2017, 09:36 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<X+.5, then our function would be (X[X+.5]), then I thought about adding up two integrals $$\int_{0}^{X+.5}(X[X+.5]),dx,$$+$$\int_{X+.5}^{20}(X[X+.5])\,dx,$$ but was unable to work through this integral. Last edited by skipjack; July 7th, 2017 at 01:44 AM. 
July 6th, 2017, 10:48 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,127 Thanks: 1106  
July 6th, 2017, 10:55 AM  #3  
Senior Member Joined: Sep 2015 From: USA Posts: 2,127 Thanks: 1106  Quote:
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, 02:34 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 01:43 AM. 
July 7th, 2017, 02:12 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,732 Thanks: 1810 
Your attempt (if X $\small\geqslant$ X + .5, etc.) was obviously incorrect.


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