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Welcome to March Calculus Madness!
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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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