Friday, March 25, 2022

March Calculus Madness Sweet Sixteen - Day 10: ∫ frac(x) dx

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Welcome to March Calculus Madness!


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frac(x) - HP Prime



frac(x):  fractional part function


Domain:  0 ≤ frac(x) < 1


∫ frac(x) dx for x = 0 to 1


According to the graph above, the area between resembles as a right triangle.


When 0≤x<1, frac(x) = x


Hence:

∫ frac(x) dx for x = 0 to x = 1


Note:  

lim a→1-  (∫ frac(x) dx for x = 0 to x = a)

= lim a→1-  (∫ x dx for x = 0 to x = a)

= lim a→1-  (∫ x dx for x = 0 to x = a)

= lim a→1-  (a^2/2 - 0)

= 1/2


What if the upper limit is less than 1? 


Let b where, 0≤x≤b<1  (b<1):


∫ frac(x) dx for x = 0 to x = b

= ∫ x dx for x = 0 to x = b

= b^2/2


Eddie 


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