For-this-problem-we-define-the-fractional-part-of-x-R-0-as-x-x-x-where-x-is-the-integer-part-of-x-i-e-the-greatest-integer-less-than-or-equal-to-x-a-Draw-the-function-x-in-a-cord

Question Number 180926 by Mastermind last updated on 19/Nov/22

$$\mathrm{For}\:\mathrm{this}\:\mathrm{problem},\:\mathrm{we}\:\mathrm{define}\:\mathrm{the}\: \\ $$$$\mathrm{fractional}\:\mathrm{part}\:\mathrm{of}\:\mathrm{x}\in\mathbb{R}_{\geqslant\mathrm{0}} \:\mathrm{as} \\ $$$$\left\{\mathrm{x}\right\}\:=\:\mathrm{x}\:−\:\lfloor\mathrm{x}\rfloor \\ $$$$\mathrm{where}\:\lfloor\mathrm{x}\rfloor\:\mathrm{is}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{part}\:\mathrm{of}\:\mathrm{x},\:\mathrm{i}.\mathrm{e} \\ $$$$\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{less}\:\mathrm{than}\:\mathrm{or}\:\mathrm{equal} \\ $$$$\mathrm{to}\:\mathrm{x}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Draw}\:\mathrm{the}\:\mathrm{function}\:\left\{\mathrm{x}\right\}\:\mathrm{in}\:\mathrm{a}\:\mathrm{cordinate} \\ $$$$\mathrm{system}\:\mathrm{for}\:\mathrm{0}\leqslant\mathrm{x}\leqslant\mathrm{3} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{A}_{\mathrm{n}} \:,\:\mathrm{under}\:\mathrm{the}\:\mathrm{graph} \\ $$$$\mathrm{of}\:\left\{\mathrm{x}\right\}\:\mathrm{between}\:\mathrm{0}\:\mathrm{and}\:\mathrm{n}\in\mathbb{N}\:\mathrm{as}\:\mathrm{given}\:\mathrm{by} \\ $$$$\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{n}} \left\{\mathrm{x}\right\}\mathrm{dx}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$

Commented by Frix last updated on 19/Nov/22

$$\mathrm{draw}\:\mathrm{it},\:\mathrm{you}'\mathrm{ll}\:\mathrm{immediately}\:\mathrm{see}\:\mathrm{the}\:\mathrm{solution}! \\ $$

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{Do}\:\mathrm{it}\:\mathrm{let}\:\mathrm{see}\:\mathrm{boss} \\ $$

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{understand}\:\mathrm{you} \\ $$$$ \\ $$

Commented by Frix last updated on 19/Nov/22

$$\left(\mathrm{a}\right)\:\mathrm{Draw}\:\mathrm{the}\:\mathrm{function}\:\left\{\mathrm{x}\right\}\:\mathrm{in}\:\mathrm{a}\:\mathrm{coordinate} \\ $$$$\mathrm{system}\:\mathrm{for}\:\mathrm{0}\leqslant\mathrm{x}\leqslant\mathrm{3} \\ $$$$\mathrm{Do}\:\mathrm{this}\:\mathrm{and}\:\mathrm{the}\:\mathrm{rest}\:\mathrm{is}\:\mathrm{obvious}. \\ $$

Commented by mr W last updated on 19/Nov/22

$${the}\:{graph}\:{of}\:{y}=\left\{{x}\right\}\:{is}\:{following}. \\ $$$${we}\:{can}\:{see}\:{that} \\ $$$$\int_{\mathrm{0}} ^{{n}} \left\{{x}\right\}{dx}={n}×\frac{\mathrm{1}}{\mathrm{2}}=\frac{{n}}{\mathrm{2}}. \\ $$

Commented by mr W last updated on 19/Nov/22

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{B},\:\mathrm{right}\:…\:\mathrm{which} \\ $$$$\mathrm{is}\:\mathrm{area}? \\ $$

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{this}\:\mathrm{is}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{A}\: \\ $$$$\mathrm{thank}\:\mathrm{you} \\ $$

Commented by MJS_new last updated on 19/Nov/22

$$\mathrm{some}\:\mathrm{love}\:\mathrm{to}\:\mathrm{think}\:\mathrm{for}\:\mathrm{themselves}, \\ $$$$\mathrm{some}\:\mathrm{love}\:\mathrm{to}\:\mathrm{let}\:\mathrm{others}\:\mathrm{think}\:\mathrm{for}\:\mathrm{them}… \\ $$$$\mathrm{guess}\:\mathrm{which}\:\mathrm{ones}\:\mathrm{might}\:\mathrm{once}\:\mathrm{reach}\:\mathrm{the}\: \\ $$$${Mastery}\:{of}\:{Mind} \\ $$

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