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

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 cordinate system for 0≤x≤3 (b) Find the area A_n , under the graph of {x} between 0 and n∈N as given by A_n =∫_0 ^n {x}dx. M.m

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

Commented by Frix last updated on 19/Nov/22

draw it, you′ll immediately see the solution!

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

Commented by Mastermind last updated on 19/Nov/22

Do it let see boss

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

Commented by Mastermind last updated on 19/Nov/22

I don′t understand you

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

Commented by Frix last updated on 19/Nov/22

(a) Draw the function {x} in a coordinate system for 0≤x≤3 Do this and the rest is obvious.

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

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

the graph of y={x} is following. we can see that ∫_0 ^n {x}dx=n×(1/2)=(n/2).

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

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

Commented by Mastermind last updated on 19/Nov/22

this is the answer to B, right ... which is area?

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

Commented by Mastermind last updated on 19/Nov/22

this is answer to A thank you

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

Commented by MJS_new last updated on 19/Nov/22

some love to think for themselves, some love to let others think for them... guess which ones might once reach the Mastery of Mind

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

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