<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/c77a182f-b0ce-43ef-b5fd-cfee72b26e54/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls10 ws8">Cálculo II</div><div class="t m0 x2 h3 y2 ff1 fs1 fc1 sc0 ls10 ws8">Curvas Polares</div><div class="t m0 x3 h4 y3 ff2 fs2 fc2 sc0 ls11 ws8">( <span class="ls10 ws0">)<span class="_0 blank"></span><span class="ls11 ws8">( <span class="ls10">)</span></span></span></div><div class="t m0 x4 h5 y4 ff3 fs3 fc2 sc0 ls0">P<span class="ls1">ara</span>me<span class="ls1">tri</span>z<span class="ls1">aç<span class="ls10 ws1">ã o </span></span><span class="ws8"> de <span class="ls1">cur</span>v<span class="ls1">as</span> e <span class="ls1">cur</span>v<span class="ls1">as</span></span><span class="ls10 wsb"> p<span class="_1 blank"> </span><span class="ls1">ol</span><span class="ws2">a r e s</span></span></div><div class="t m0 x5 h6 y5 ff4 fs3 fc2 sc0 ls1">Parametr<span class="ls0">iz</span>ar<span class="ls10 wsb"> u<span class="_1 blank"> </span></span>ma<span class="ls0 ws8"> curva é </span>escre<span class="ls0">v</span>er<span class="ls0 ws8"> </span>os<span class="ls0 ws8"> pon</span>tos<span class="ls10 wsb"> d</span>e<span class="ls10 wsb"> u<span class="_1 blank"> </span></span>ma<span class="ls0 ws8"> curva </span>em<span class="ls0 ws8"> fun</span>ção<span class="ls10 wsb"> d<span class="_1 blank"> </span><span class="ws8">e<span class="_1 blank"> </span> </span></span></div><div class="t m0 x5 h6 y6 ff4 fs3 fc2 sc0 ls10 ws1">u m <span class="wsb"> p<span class="_1 blank"> </span><span class="ls1">arâmetr</span><span class="ws8">o<span class="_1 blank"> </span> <span class="_1 blank"> </span><span class="ff5 ls0">t</span><span class="ff6">.</span></span></span></div><div class="t m0 x5 h7 y7 ff2 fs4 fc3 sc0 ls10">{</div><div class="t m0 x5 h6 y8 ff4 fs3 fc2 sc0 ls1">Lem<span class="ls0">b</span>re<span class="ls0">-</span>se<span class="ls10 wsb"> d<span class="_1 blank"> </span></span>e<span class="ls0 ws8"> indi</span>car<span class="ws8"> em<span class="ls0"> qu</span>e<span class="ls0"> in</span>ter<span class="ls10">v<span class="_1 blank"> </span>a<span class="_1 blank"> </span>l<span class="_1 blank"> </span>o <span class="_1 blank"> </span>o<span class="_1 blank"> </span><span class="wsb"> p<span class="_1 blank"> </span></span></span>arâmetro<span class="ls2"> </span><span class="ff5">t</span><span class="ls0"> </span>está</span><span class="ls10 wsb"> d<span class="_1 blank"> </span></span>e<span class="ff7 ls0">\ufb01nid</span><span class="ls10 ws3">o .</span></div><div class="t m0 x5 h8 y9 ff1 fs3 fc3 sc0 ls10 ws1">( I ) <span class="fc2 wsb"> D<span class="_1 blank"> </span><span class="ls1">er<span class="ls0">i</span>vada<span class="ls0 ws8"> </span>"carte<span class="ls0">si</span></span><span class="ws1">a n <span class="ls1">a"<span class="ls0 ws8"> </span>de</span></span> u<span class="_1 blank"> </span><span class="ls1">ma<span class="ls0 ws8"> cu</span>rva<span class="ls0 ws8"> </span>parametr<span class="ls3">i</span>zada</span>:</span></div><div class="t m0 x5 h5 ya ff3 fs3 fc2 sc0 ls10 wsb"> <span class="ws8"> </span></div><div class="t m0 x5 h5 yb ff3 fs3 fc2 sc0 ls10 wsb"> <span class="ws8"> <span class="_1 blank"> </span>=</span></div><div class="t m0 x5 h8 yc ff1 fs3 fc3 sc0 ls0">(<span class="ls1">II<span class="ls10 ws8">)<span class="_1 blank"> </span> <span class="_1 blank"> </span></span></span><span class="fc2">Co<span class="ls1">mpr</span>i<span class="ls1">me<span class="ls10 ws1">n t </span></span><span class="wsa">o <span class="_1 blank"> </span><span class="ls1">de</span></span><span class="ls10 wsb"> u<span class="_1 blank"> </span><span class="ls1">ma</span></span><span class="ws8"> cu<span class="ls1">rva</span> <span class="ls1">parametr<span class="ls3">i</span>zada<span class="ls10">:</span></span></span></span></div><div class="t m0 x5 h5 yd ff5 fs3 fc2 sc0 ls1">L<span class="ls0 ws8"> = + d<span class="ls10">t</span></span></div><div class="t m0 x5 h8 ye ff1 fs3 fc3 sc0 ls0">(<span class="ls1">III<span class="ls10 ws8">)<span class="_1 blank"> </span> <span class="_1 blank"> </span></span><span class="fc2">Área<span class="ls0 ws8"> </span>de<span class="ls0 ws8"> cu</span>rva<span class="ls0 ws8"> </span>parametr<span class="ls3">i</span>zada<span class="ls10">:</span></span></span></div><div class="t m0 x5 h5 yf ff5 fs3 fc2 sc0 ls0 ws8">A = <span class="ls4">y<span class="ls10 wsb"> d<span class="_1 blank"> </span>t</span></span></div><div class="t m0 x5 h9 y10 ff4 fs3 fc2 sc0 ls1">Em<span class="ls5 ws8"> <span class="ff5 ls0">c<span class="ls1">oor</span>denad<span class="ls1">as<span class="ls10 wsb"> p<span class="_1 blank"> </span><span class="ws1">o l </span></span>ar<span class="ls10 ws2">e s </span></span></span></span><span class="ls10 wsb"> o<span class="_1 blank"> </span></span>s<span class="ls10 wsb"> p<span class="_1 blank"> </span><span class="ws8">o<span class="_1 blank"> </span>n<span class="_1 blank"> </span>t<span class="_1 blank"> </span>o<span class="_1 blank"> </span>s<span class="_1 blank"> </span> <span class="_1 blank"> </span></span></span>são<span class="ls6 ws8"> </span><span class="ff5">P<span class="ls0 ws8"> = </span>(r;<span class="ff8 ls0 wsa"> <span class="_1 blank"> </span>\u03b8</span><span class="ls7">)</span></span><span class="ws8">, o<span class="ls0">nd<span class="ls10">e<span class="_1 blank"> </span> <span class="_1 blank"> </span></span></span><span class="ff5">r</span><span class="ls10"> <span class="_1 blank"> </span>é <span class="_1 blank"> </span>a<span class="_1 blank"> </span><span class="ls0"> di</span></span>stâ</span><span class="ls10 ws1">n c i a </span><span class="ls0 ws8"> do<span class="ls10"> </span></span></div><div class="t m0 x5 h9 y11 ff4 fs3 fc2 sc0 ls10 ws8">p<span class="_1 blank"> </span>o<span class="_1 blank"> </span>n<span class="_1 blank"> </span>to<span class="_1 blank"> </span> <span class="_1 blank"> </span><span class="ff5 ls0">P</span> <span class="_1 blank"> </span>à<span class="_1 blank"> </span><span class="wsb"> o<span class="_1 blank"> </span><span class="ls1">r<span class="ls0">ig</span>em</span></span><span class="ls0"> do pl</span><span class="ws3">a n o <span class="wsb"> p<span class="_1 blank"> </span><span class="ws1">o l <span class="ls1">ar,</span></span></span></span> <span class="_1 blank"> </span>e<span class="_1 blank"> </span> <span class="_2 blank"> </span><span class="ff8 ls8">\u03b8</span> <span class="_1 blank"> </span>é<span class="_1 blank"> </span><span class="ls0"> o <span class="ls1">â</span>ngul<span class="ls8">o</span> qu</span>e <span class="_1 blank"> </span>o<span class="_1 blank"> </span> <span class="_1 blank"> </span><span class="ls1">se<span class="ls0">g</span>me</span>n<span class="_1 blank"> </span>t<span class="_1 blank"> </span>o<span class="_1 blank"> </span> <span class="_2 blank"> </span><span class="ff5 ws1">O P </span><span class="wsb"> f<span class="_1 blank"> </span></span>a<span class="_1 blank"> </span>z<span class="_1 blank"> </span> </div><div class="t m0 x5 h6 y12 ff4 fs3 fc2 sc0 ls1">com<span class="ls10 ws8"> <span class="_1 blank"> </span>o<span class="_1 blank"> </span> <span class="_1 blank"> </span>e<span class="ls0">ix</span></span>o<span class="ls10 wsb"> p<span class="_1 blank"> </span><span class="ws1">o l </span></span>ar<span class="ls10 wsb"> n<span class="_1 blank"> </span><span class="ws8">o<span class="_1 blank"> </span> <span class="_1 blank"> </span></span></span>se<span class="ls10 ws1">n t <span class="ls0">id</span></span>o<span class="ls10 wsb"> h<span class="_1 blank"> </span></span>orár<span class="ls0">i</span>o.<span class="ls0 ws8"> A </span>represe<span class="ls0">n</span>tação<span class="ls10 wsb"> p<span class="_1 blank"> </span><span class="ws1">o l </span></span>ar<span class="ls10 wsb"> d<span class="_1 blank"> </span><span class="ws8">e<span class="_1 blank"> </span> <span class="_1 blank"> </span>c<span class="_3 blank"> </span>u<span class="_1 blank"> </span>r<span class="_1 blank"> </span>v<span class="_1 blank"> </span></span></span>as<span class="ls10 ws8"> <span class="_1 blank"> </span>é </span></div><div class="t m0 x5 h6 y13 ff4 fs3 fc2 sc0 ls0 ws4">fei<span class="ls1">ta<span class="ls10 wsb"> u<span class="_1 blank"> </span></span>sa</span><span class="wsa">ndo <span class="_1 blank"> </span><span class="ls1">as</span><span class="ws8"> <span class="ls1">se</span>guin<span class="ls1">tes<span class="ls10 wsb"> p<span class="_1 blank"> </span></span>arametr</span>iz<span class="ls1">aç<span class="ls3">õ</span>es<span class="ls10">:</span></span></span></span></div><div class="t m0 x5 ha y14 ff5 fs3 fc2 sc0 ls9">x<span class="fs5 lsa v1">2</span><span class="ls12 ws8 v0"> + <span class="lsb">y<span class="fs5 lsa v1">2</span></span> = <span class="lsc">r<span class="fs5 ls10 v1">2<span class="_1 blank"> </span> </span></span></span></div><div class="t m0 x5 h9 y15 ff4 fs3 fc2 sc0 ls1">ta<span class="ls10 ws8">n<span class="_1 blank"> </span> <span class="_1 blank"> </span><span class="ff8 ls0">\u03b8</span> <span class="_1 blank"> </span>=</span></div><div class="t m0 x6 hb y16 ff5 fs6 fc2 sc0 ls10 ws5">d x</div><div class="t m0 x7 hb y17 ff5 fs6 fc2 sc0 ls10 ws5">d y</div><div class="t m0 x8 hc y18 ff5 fs7 fc2 sc0 ls10">x</div><div class="t m0 x8 hc y19 ff5 fs7 fc2 sc0 ls10">y</div><div class="t m0 x9 h5 y1a ff5 fs3 fc2 sc0 lsd">y<span class="ls13 ws8"> = </span>y(t<span class="ls10">)</span></div><div class="t m0 x9 h5 y1b ff5 fs3 fc2 sc0 ls13 ws8">x = x<span class="lsd">(t<span class="ls10">)</span></span></div><div class="t m0 xa h9 y1c ff5 fs3 fc2 sc0 lsd">y<span class="ls13 ws8"> = <span class="ls10">r<span class="_4 blank"> </span> <span class="_4 blank"> </span>s<span class="_4 blank"> </span>i<span class="_4 blank"> </span>n<span class="_4 blank"> </span><span class="ff8">\u03b8</span></span></span></div><div class="t m0 xa h9 y1d ff5 fs3 fc2 sc0 ls13 ws8">x = <span class="lse">r<span class="ls10 wsd"> c<span class="_4 blank"> </span><span class="ws6">o s <span class="ff8 ls1">\u03b8</span><span class="ws8"> </span></span></span></span></div><div class="t m0 xb hd y1e ff5 fs8 fc2 sc0 ls10 ws7">d t</div><div class="t m0 xb hd y1f ff5 fs8 fc2 sc0 ls10 ws7">d t</div><div class="t m0 xb hd y20 ff5 fs8 fc2 sc0 ls10 ws7">d y</div><div class="t m0 xc hd y21 ff5 fs8 fc2 sc0 ls10 ws7">d x</div><div class="t m0 xd hd y22 ff5 fs8 fc2 sc0 ls10 ws7">d t</div><div class="t m0 xd hd y23 ff5 fs8 fc2 sc0 ls10 ws7">d x</div><div class="t m0 xe hd y24 ff5 fs8 fc2 sc0 ls10">a</div><div class="t m0 xf hd y25 ff5 fs8 fc2 sc0 lsf">b<span class="fs9 ls10 v2">2</span></div><div class="t m0 x10 h4 y26 ff2 fs2 fc2 sc0 ls11 ws8">( <span class="ls10">)</span></div><div class="t m0 x11 hd y27 ff5 fs8 fc2 sc0 ls10 ws7">d t</div><div class="t m0 x11 hd y28 ff5 fs8 fc2 sc0 ls10 ws7">d x</div><div class="t m0 x12 he y29 ff5 fs9 fc2 sc0 ls10">2</div><div class="t m0 x13 hd y22 ff5 fs8 fc2 sc0 ls10 ws8">d<span class="_2 blank"> </span>t<span class="_3 blank"> </span> <span class="_3 blank"> </span> </div><div class="t m0 x13 hd y23 ff5 fs8 fc2 sc0 ls10 ws7">d y</div><div class="t m0 xe hd y2a ff5 fs8 fc2 sc0 ls10">a</div><div class="t m0 xf hd y2b ff5 fs8 fc2 sc0 ls10">b</div><div class="t m0 x14 h8 y2c ff1 fs3 fc0 sc0 ls10">1</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/c77a182f-b0ce-43ef-b5fd-cfee72b26e54/bg2.png"><div class="t m0 x5 h8 y2d ff9 fs3 fc3 sc0 ls10 ws1">( I ) <span class="fc2 wsb"> D<span class="_1 blank"> </span><span class="ls1">er<span class="ls0">i</span>vada<span class="ls0 ws8"> </span>"carte<span class="ls0">si</span></span><span class="ws1">a n <span class="ls1">a"<span class="ls0 ws8"> </span>de</span></span> u<span class="_1 blank"> </span><span class="ls1">ma<span class="ls0 ws8"> cu</span>rva</span><span class="ws8"> <span class="_1 blank"> </span>p<span class="_1 blank"> </span><span class="ls0">ol<span class="ls1">ar</span></span>:</span></span></div><div class="t m0 x5 h8 y2e ff9 fs3 fc2 sc0 ls10 ws8"> </div><div class="t m0 x5 h8 y2f ff9 fs3 fc2 sc0 ls0 ws8"> = =<span class="ls10"> </span></div><div class="t m0 x5 h8 y30 ff9 fs3 fc3 sc0 ls0">(<span class="ls1">II<span class="ls5">)</span></span><span class="fc2 ws8"> Co<span class="ls1">mprime<span class="ls10 ws1">n t </span></span><span class="wsa">o <span class="_1 blank"> </span><span class="ls1">de<span class="ls10 wsb"> u<span class="_1 blank"> </span></span>ma</span></span> cu<span class="ls1">rva<span class="ls10"> <span class="_1 blank"> </span>p<span class="_1 blank"> </span></span></span>ol<span class="ls1">ar<span class="ls10">:</span></span></span></div><div class="t m0 x5 hf y31 ffa fs3 fc2 sc0 ls1">L<span class="ls0 ws8"> = </span>(r<span class="ls14">)<span class="fs5 ls15 v1">2</span><span class="ls0 ws8 v0"> + d<span class="ffb ls10">\u03b8</span></span></span></div><div class="t m0 x5 h8 y32 ff9 fs3 fc3 sc0 ls0">(<span class="ls1">III<span class="ls16">)</span></span><span class="fc2 ws8"> <span class="ls1">Área</span> <span class="ls1">de</span> cu<span class="ls1">rva<span class="ls10"> <span class="_1 blank"> </span>p<span class="_1 blank"> </span></span></span>ol<span class="ls1">ar<span class="ls10">:</span></span></span></div><div class="t m0 x5 h10 y33 ffa fs3 fc2 sc0 ls0 ws8">A = <span class="ls17">r<span class="fs5 ls18 v1">2</span></span><span class="v0"> d<span class="ffb ls10">\u03b8</span></span></div><div class="t m0 x6 hc y34 ffa fs7 fc2 sc0 ls10 wse">d x</div><div class="t m0 x6 hc y35 ffa fs7 fc2 sc0 ls10 wse">d y</div><div class="t m0 x15 hc y36 ffa fs7 fc2 sc0 ls10">2</div><div class="t m0 x15 hc y37 ffa fs7 fc2 sc0 ls10">1</div><div class="t m0 x3 h11 y38 ffa fsa fc2 sc0 ls19">d<span class="ffb ls10">\u03b8</span></div><div class="t m0 x3 h12 y39 ffa fsa fc2 sc0 ls10 wsf">d r</div><div class="t m0 xc h12 y3a ffa fsa fc2 sc0 ls10 wsf">d t</div><div class="t m0 xb h12 y3b ffa fsa fc2 sc0 ls10 wsf">d y</div><div class="t m0 xc h12 y3c ffa fsa fc2 sc0 ls10 wsf">d t</div><div class="t m0 xc h12 y3d ffa fsa fc2 sc0 ls10 wsf">d x</div><div class="t m0 x16 h13 y3e ffa fs7 fc2 sc0 ls10 ws10">s i n <span class="ffb ls1d ws11">\u03b8 <span class="_1 blank"> </span>+ </span><span class="ls1a">r</span><span class="ws12"> c<span class="_1 blank"> </span><span class="ls1a">os</span><span class="ffb">\u03b8</span></span></div><div class="t m0 x3 h11 y3f ffa fsa fc2 sc0 ls19">d<span class="ffb ls10">\u03b8</span></div><div class="t m0 x3 h12 y40 ffa fsa fc2 sc0 ls10 wsf">d r<span class="_5 blank"> </span><span class="fs7 ws10 v3">c o s <span class="ffb ls1d ws11">\u03b8 <span class="_1 blank"> </span>+ <span class="ffa ls1a ws8">r s<span class="ls1d">in</span></span><span class="ls10">\u03b8</span></span></span></div><div class="t m0 x17 h4 y41 ffc fs2 fc2 sc0 ls11 ws8">( <span class="ls10">)</span></div><div class="t m0 xe h14 y42 ffb fs8 fc2 sc0 ls10">\u03b1</div><div class="t m0 xf h14 y43 ffb fs8 fc2 sc0 ls10">\u03b2</div><div class="t m0 x18 h14 y44 ffb fs8 fc2 sc0 ls10">\u03b1</div><div class="t m0 x19 h14 y45 ffb fs8 fc2 sc0 ls10">\u03b2</div><div class="t m0 x1a he y46 ffa fs9 fc2 sc0 ls10">2</div><div class="t m0 x1b hd y47 ffa fs8 fc2 sc0 ls1b">d<span class="ffb fsa ls1c">\u03b8</span><span class="ls10 ws8"> <span class="_2 blank"> </span> </span></div><div class="t m0 x1b hd y48 ffa fs8 fc2 sc0 ls10 ws7">d r</div><div class="t m0 x14 h8 y2c ff9 fs3 fc0 sc0 ls10">2</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>