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August 17th, 2019, 06:35 AM   #1
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Definite integral of box function

Evaluate integral if [ t + 1]^3 dt from t = 0 to t=x where [.] denotes the box function

Please let me know how to proceed
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August 17th, 2019, 08:02 AM   #2
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You might find it useful to sketch the graph of the function.
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August 17th, 2019, 01:47 PM   #3
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You are integrating a function which is piecewise constant.
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August 17th, 2019, 10:47 PM   #4
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I proceeded as below:

1. break down interval [ 0 , x] in x sub-intervals [ 0,1 ] , [ 1,2] .... [x-1,x]

then integrate

2. It comes to 1^3 + 2^3 ... [x]^3 = (([x]*([x) +1))/2)^2

But the answer is not matching. Need to know - where I am going wrong?

Last edited by skipjack; August 18th, 2019 at 02:09 PM.
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August 18th, 2019, 01:35 PM   #5
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I assume x is an integer. If so, your answer looks correct. What is the given answer? If x is not an integer, the last term is incorrect. It should be $[x]^3(x-[x])$.
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August 18th, 2019, 02:47 PM   #6
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If x isn't an integer, it comes to 1³ + 2³ + . . . + [x]³ + ([[x] + 1)³(x - [x]).
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August 18th, 2019, 11:41 PM   #7
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Many thanks for your help. Answer from skipjack is matching. Yes, nowhere is it mentioned that x is an integer.

However, could you please explain how the the last term is coming ([[x] + 1)³(x - [x])?

Last edited by skipjack; August 19th, 2019 at 09:30 AM.
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August 19th, 2019, 05:34 AM   #8
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For t between [x] and x, [t + 1]³ = ([x] + 1)³.
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August 19th, 2019, 08:37 AM   #9
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Many thanks. But I still have a question - how to prove analytically

integral ([t] + 1)^3 from [x] to x is ([x] + 1)^3 (x - [x]).

Last edited by skipjack; August 19th, 2019 at 09:18 AM.
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August 19th, 2019, 09:27 AM   #10
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For [x] < t < x, [t] = [x] (a constant), so ([t] + 1)³ is also a constant.
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