Um ponto material móvel P(-2 + t, 4t/3 + 2) desloca-se no plano cartesiano e suas coordenadas variam em função do tempo t (t ≥0). A distância percorrida pelo ponto material móvel entre o ponto A para t = 0 e o ponto B para t = 6, é
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O movimento do ponto é descrito pela curva
Achando o vetor velocidade da curva:
![\vec{\varphi'(t)}=(x'(t),~y'(t))\\\\\\\vec{\varphi'(t)}=\left(\dfrac{d}{dt}\bigg[-2+t\bigg],\dfrac{d}{dt}\left[\dfrac{4}{3}t+2\right]\right)\\\\\\\boxed{\boxed{\vec{\varphi'(t)}=\left(1,~\frac{4}{3}\right)}}](https://tex.z-dn.net/?f=%5Cvec%7B%5Cvarphi%27%28t%29%7D%3D%28x%27%28t%29%2C%7Ey%27%28t%29%29%5C%5C%5C%5C%5C%5C%5Cvec%7B%5Cvarphi%27%28t%29%7D%3D%5Cleft%28%5Cdfrac%7Bd%7D%7Bdt%7D%5Cbigg%5B-2%2Bt%5Cbigg%5D%2C%5Cdfrac%7Bd%7D%7Bdt%7D%5Cleft%5B%5Cdfrac%7B4%7D%7B3%7Dt%2B2%5Cright%5D%5Cright%29%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cboxed%7B%5Cvec%7B%5Cvarphi%27%28t%29%7D%3D%5Cleft%281%2C%7E%5Cfrac%7B4%7D%7B3%7D%5Cright%29%7D%7D "\vec{\varphi'(t)}=(x'(t),~y'(t))\\\\\\\vec{\varphi'(t)}=\left(\dfrac{d}{dt}\bigg[-2+t\bigg],\dfrac{d}{dt}\left[\dfrac{4}{3}t+2\right]\right)\\\\\\\boxed{\boxed{\vec{\varphi'(t)}=\left(1,~\frac{4}{3}\right)}}")
O comprimento da curva entre o ponto A e o ponto B é dado pela integral
![L=\displaystyle\int\limits_{0}^{6}\vec{||\varphi'(t)||}dt\\\\\\L=\int\limits_{0}^{6}\sqrt{[x'(t)]^{2}+[y'(t)]^{2}}dt\\\\\\L=\int\limits_{0}^{6}\sqrt{1^{2}+\left[\dfrac{4}{3}\right]^{2}}dt\\\\\\L=\int\limits_{0}^{6}\sqrt{1+\dfrac{16}{9}}dt](https://tex.z-dn.net/?f=L%3D%5Cdisplaystyle%5Cint%5Climits_%7B0%7D%5E%7B6%7D%5Cvec%7B%7C%7C%5Cvarphi%27%28t%29%7C%7C%7Ddt%5C%5C%5C%5C%5C%5CL%3D%5Cint%5Climits_%7B0%7D%5E%7B6%7D%5Csqrt%7B%5Bx%27%28t%29%5D%5E%7B2%7D%2B%5By%27%28t%29%5D%5E%7B2%7D%7Ddt%5C%5C%5C%5C%5C%5CL%3D%5Cint%5Climits_%7B0%7D%5E%7B6%7D%5Csqrt%7B1%5E%7B2%7D%2B%5Cleft%5B%5Cdfrac%7B4%7D%7B3%7D%5Cright%5D%5E%7B2%7D%7Ddt%5C%5C%5C%5C%5C%5CL%3D%5Cint%5Climits_%7B0%7D%5E%7B6%7D%5Csqrt%7B1%2B%5Cdfrac%7B16%7D%7B9%7D%7Ddt "L=\displaystyle\int\limits_{0}^{6}\vec{||\varphi'(t)||}dt\\\\\\L=\int\limits_{0}^{6}\sqrt{[x'(t)]^{2}+[y'(t)]^{2}}dt\\\\\\L=\int\limits_{0}^{6}\sqrt{1^{2}+\left[\dfrac{4}{3}\right]^{2}}dt\\\\\\L=\int\limits_{0}^{6}\sqrt{1+\dfrac{16}{9}}dt")
![L=\displaystyle\int\limits_{0}^{6}\sqrt{\dfrac{25}{9}}dt\\\\\\L=\int\limits_{0}^{6}\dfrac{5}{3}dt\\\\\\L=\dfrac{5}{3}\int\limits_{0}^{6}dt\\\\\\L=\dfrac{5}{3}[~t~]_{0}^{6}\\\\\\L=\dfrac{5}{3}\cdot(6-0)\\\\\\L=\dfrac{5}{3}\cdot6=5\cdot2\\\\\\\boxed{\boxed{L=10~u.c}}](https://tex.z-dn.net/?f=L%3D%5Cdisplaystyle%5Cint%5Climits_%7B0%7D%5E%7B6%7D%5Csqrt%7B%5Cdfrac%7B25%7D%7B9%7D%7Ddt%5C%5C%5C%5C%5C%5CL%3D%5Cint%5Climits_%7B0%7D%5E%7B6%7D%5Cdfrac%7B5%7D%7B3%7Ddt%5C%5C%5C%5C%5C%5CL%3D%5Cdfrac%7B5%7D%7B3%7D%5Cint%5Climits_%7B0%7D%5E%7B6%7Ddt%5C%5C%5C%5C%5C%5CL%3D%5Cdfrac%7B5%7D%7B3%7D%5B%7Et%7E%5D_%7B0%7D%5E%7B6%7D%5C%5C%5C%5C%5C%5CL%3D%5Cdfrac%7B5%7D%7B3%7D%5Ccdot%286-0%29%5C%5C%5C%5C%5C%5CL%3D%5Cdfrac%7B5%7D%7B3%7D%5Ccdot6%3D5%5Ccdot2%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cboxed%7BL%3D10%7Eu.c%7D%7D "L=\displaystyle\int\limits_{0}^{6}\sqrt{\dfrac{25}{9}}dt\\\\\\L=\int\limits_{0}^{6}\dfrac{5}{3}dt\\\\\\L=\dfrac{5}{3}\int\limits_{0}^{6}dt\\\\\\L=\dfrac{5}{3}[~t~]_{0}^{6}\\\\\\L=\dfrac{5}{3}\cdot(6-0)\\\\\\L=\dfrac{5}{3}\cdot6=5\cdot2\\\\\\\boxed{\boxed{L=10~u.c}}")
Achando o vetor velocidade da curva:
O comprimento da curva entre o ponto A e o ponto B é dado pela integral
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