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Evaluate the line integral, where c is the given curve. ∫c zdx + xdy + ydz
c.x = t3, y = t4, z = t3, 0 ≤ t ≤ 1evaluate the line integral, where c is the given curve. ∫c zdx + xdy + ydz
c.x = t3, y = t4, z = t3, 0 ≤ t ≤ 1

Respuesta :

LammettHash
[tex]\displaystyle\int_{\mathcal C}z\,\mathrm dx+x\,\mathrm dy+y\,\mathrm dz[/tex]

[tex]x=t^3\implies\mathrm dx=3t^2\,\mathrm dt[/tex]
[tex]y=t^4\implies\mathrm dy=4t^3\,\mathrm dt[/tex]
[tex]z=t^3\implies\mathrm dz=3t^2\,\mathrm dt[/tex]

So the integral is equivalent to

[tex]\displaystyle\int_{t=0}^{t=1}(3t^2+4t^3+3t^2)\,\mathrm dt=3[/tex]

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