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Definite Integrals
The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.
\[ \displaystyle \int_{0}^{20} \Bigl(\lfloor x \rfloor \{x\} \Bigr) \ dx = \ ? \]
Details and assumptions:
Every \(x\in \mathbb{R}\) can be written as \(x=\lfloor x \rfloor + \{x\} \).
\(\lfloor x \rfloor\) denotes greatest integer less than or equal to \(x\).
\(\{x\} \) is the fractional part of \(x\).
by
Aditya Raut
by
Michael Tong
Let \(S_N\) satisfy the equation above. What is the value of \(\displaystyle \lim _{ n\rightarrow \infty }{ \left( { S }_{ 2n }-{ S }_{ n } \right) }\)?
Give your answer to three decimal places.
by
Atul Antony Zachariahs
\( f(x) = \begin{cases}{1-|x|}, && {|x|\>\le\>1} \\ {|x|-1,} && {|x|>1}\end{cases} \)
\(g(x) = f(x-1)+f(x+1)\).
Given the two functions above, what is the value of \( \displaystyle \int _{ -3 }^{ 5 }{ g(x) \mathrm{d}x } \)?
by
Harshvardhan Mehta
\[\large \int_0^{\pi /2} ( \sin^{2014}x - \cos^{2014}x ) \,dx = \, ? \]
by
Soham Zemse