Question

Resolvendo a Integral?
Foto em anexo.

Answer 1

Calcular a integral de linha de um campo vetorial sobre a curva $C:$

$\displaystyle\int_C y\,dx+x\,dy$

sendo $\overrightarrow{\mathbf{F}}(x,\,y)=(y,\,x)$ o campo vetorial

e a curva $C$ é o segmento de reta

$y=x~~~~~\text{com }0\le x\le 1.$

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Parametrizando a curva:

$C:~\left\{ \!\begin{array}{l} x=t\\ y=t \end{array} \right.~~~~~~0\le t\le 1$

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Vetor tangente à curva:

$C(t)=(t,\,t)\\\\ C'(t)=(1,\,1)$

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Sendo assim, a integral de linha é

$\displaystyle\int_C \overrightarrow{\mathbf{F}}(x,\,y)\cdot d\overrightarrow{\mathbf{r}}\\\\\\ =\int_{0}^{1} \overrightarrow{\mathbf{F}}(t,\,t)\cdot C'(t)\,dt\\\\\\ =\int_{0}^{1} (t,\,t)\cdot (1,\,1)\,dt\\\\\\ =\int_{0}^{1} (t\cdot 1+t\cdot 1)\,dt\\\\\\ =\int_{0}^{1} 2t\,dt\\\\\\ =t^2\big|_0^1\\\\ =1^2-0^2\\\\ =1$


Resposta: alternativa e. 1.