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Exercise Given a, b ∈ R^∗ and t is a variable real. 1) Solve in R^2 for x,y this system: { ((xsin t−ycos t=−a)),((xcos t+ysin t=b.)) :} 2)/Demonstrate that these solutions can be written like this ( r and θ ∈ R). { ((x=rcos(t+θ))),((y=rsin(t+θ))) :} 3) we suppose now that a=b=1 and θ=(π/(12)) solve this in [0;2π[ { ((rcos(t+θ)≥−1)),((rsin(t+θ)<−1)) :}

$$\mathrm{Exercise} \\ $$$$\mathrm{Given}\:\mathrm{a},\:\mathrm{b}\:\in\:\mathbb{R}^{\ast} \:\mathrm{and}\:{t}\:\mathrm{is}\:\mathrm{a}\:\mathrm{variable}\:\mathrm{real}. \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Solve}\:\mathrm{in}\:\mathbb{R}^{\mathrm{2}} \:\mathrm{for}\:{x},{y}\:\mathrm{this}\:\mathrm{system}:\: \\ $$$$\begin{cases}{{x}\mathrm{sin}\:{t}−{y}\mathrm{cos}\:{t}=−\mathrm{a}}\\{{x}\mathrm{cos}\:{t}+{y}\mathrm{sin}\:{t}={b}.}\end{cases} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)/\mathrm{Demonstrate}\:\mathrm{that}\:\mathrm{these}\:\mathrm{solutions}\:\mathrm{can} \\ $$$$\mathrm{be}\:\mathrm{written}\:\mathrm{like}\:\mathrm{this}\:\left(\:{r}\:\mathrm{and}\:\theta\:\in\:\mathbb{R}\right). \\ $$$$\begin{cases}{{x}={rcos}\left({t}+\theta\right)}\\{{y}={rsin}\left({t}+\theta\right)}\end{cases} \\ $$$$ \\ $$$$\left.\mathrm{3}\right)\:\mathrm{we}\:\mathrm{suppose}\:\mathrm{now}\:\mathrm{that}\:\mathrm{a}=\mathrm{b}=\mathrm{1}\:\mathrm{and}\:\theta=\frac{\pi}{\mathrm{12}} \\ $$$$\mathrm{solve}\:\mathrm{this}\:\mathrm{in}\:\left[\mathrm{0};\mathrm{2}\pi\left[\right.\right. \\ $$$$\begin{cases}{{rcos}\left({t}+\theta\right)\geqslant−\mathrm{1}}\\{{rsin}\left({t}+\theta\right)<−\mathrm{1}}\end{cases} \\ $$

Question Number 94997 Answers: 2 Comments: 0

Question Number 94992 Answers: 4 Comments: 0

1. Show that: ∫_0 ^( π) ((xdx)/((a^2 sin^2 x+b^2 cos^2 x)^2 )) = ((π^2 (a^2 +b^2 ))/(4a^3 b^3 )) 2.The density at the point (x,y) of a lamina bounded by the circle x^2 +y^2 −2ax=0 is ϱ =x find its mass. 3.∗ 4. If z= ((cos y)/x) and x=u^2 −v , y=e^x find (dz/dv).

$$\:\mathrm{1}.\:\mathrm{Show}\:\:\mathrm{that}: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\:\pi} \:\:\:\frac{\mathrm{xdx}}{\left(\mathrm{a}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \mathrm{x}+\mathrm{b}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \mathrm{x}\right)^{\mathrm{2}} }\:=\:\frac{\pi^{\mathrm{2}} \left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \right)}{\mathrm{4a}^{\mathrm{3}} \mathrm{b}^{\mathrm{3}} } \\ $$$$\:\: \\ $$$$\:\mathrm{2}.\mathrm{The}\:\mathrm{density}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{of}\:\mathrm{a}\:\mathrm{lamina}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\:\:\:\:\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{2ax}=\mathrm{0}\:\mathrm{is}\:\varrho\:=\mathrm{x}\:\:\:\mathrm{find}\:\mathrm{its}\:\mathrm{mass}. \\ $$$$\:\mathrm{3}.\ast \\ $$$$\:\mathrm{4}.\:\mathrm{If}\:\:\mathrm{z}=\:\frac{\mathrm{cos}\:\mathrm{y}}{\mathrm{x}}\:\mathrm{and}\:\mathrm{x}=\mathrm{u}^{\mathrm{2}} −\mathrm{v}\:,\:\mathrm{y}=\mathrm{e}^{\mathrm{x}} \:\mathrm{find}\:\frac{\mathrm{dz}}{\mathrm{dv}}. \\ $$$$\:\: \\ $$

Question Number 94984 Answers: 1 Comments: 2

(d/dx) [(1/x) +(√x) ] ?

$$\frac{\mathrm{d}}{\mathrm{dx}}\:\left[\frac{\mathrm{1}}{\mathrm{x}}\:+\sqrt{\mathrm{x}}\:\right]\:? \\ $$

Question Number 94969 Answers: 1 Comments: 0

y′′−3y′+2y=0 ; y=0,y′=3 for x=0

$$\mathrm{y}''−\mathrm{3y}'+\mathrm{2y}=\mathrm{0}\:;\:\mathrm{y}=\mathrm{0},\mathrm{y}'=\mathrm{3}\:\mathrm{for}\:\mathrm{x}=\mathrm{0} \\ $$

Question Number 94960 Answers: 2 Comments: 0