Respuesta :
Answer:
[tex]$ \frac{2187 \pi}{4}$[/tex]
Step-by-step explanation:
Using Stoke's theorem, we get
[tex]$ \int \int_S curl F.dS = \int_C F.dr$[/tex]
Now parametrize C
r(t) = ( 3 cos t, 3 - 3 sin t ) where, t in [0 2π]
[tex]$ \int_C F. dr= \int_0 ^{2 \pi}F ( 3 \cos t, 3 - 3 \sin t). r'(t)dt$[/tex]
[tex]$= \int_0^{2 \pi}(( 9 \cos^2 t)(27)(-3\sin t), \sin t \times (-27 \cos t \sin t)) . (-3 \sin t, 0 - 3 \cos t) dt $[/tex]
[tex]$ = 27 \int_0^{2 \pi} 81 \cos ^2 t \sin ^2 t + 3 \cos ^2 t \sin tdt $[/tex]
[tex]$ = 27 \int_0^{2 \pi} \frac{81}{4} \sin ^2 2t+27 [ - \cos^3 t]_0^{2 \pi}$[/tex]
[tex]$ = 27 \times \frac{81}{4} \int_0^{2 \pi} \frac{(1- \cos 4t)}{2} $[/tex]
[tex]$= \frac{2187}{8}[t-\frac{1}{4} \sin 4t]_0^{2 \pi} $[/tex]
[tex]$= \frac{2187 \pi}{4}$[/tex]
By using Stokes Theorem the required S curl F.ds is [tex]\dfrac{2187\pi }{4}[/tex].
Given that,
Function F(x, y, z) = (x^2y^3z)i + sin(xyz)j + xyzk,
S is the part of the cone y^2 = x^2 + z^2 that lies between the planes y = 0 and y =3, oriented in the direction of the positive y-axis.
We have to determine,
Use Stokes' Theorem to evaluate S curl F·dS.
According to the question,
By using Stokes Theorem to evaluate S curl F .dS.
[tex]\int\ \int _s curlF.ds = \int _c F.dr\\\\[/tex]
Now parametrize C
r(t) = ( 3 cos t, 3 - 3 sin t ) ,
Where, t in [0 2π]
Then,
[tex]\int_c F.dr = \int^{2\pi }_0 F(3cost, 3-3sint).r'(t)dt\\\\[/tex]
[tex]= \int^{2\pi }_0 ((9cos^2t)(27)(-3sint),(sint) \times (-27cost.sint). (-3sint,0-3cost)dt\\\\= 27 \int^{2\pi }_0 81cos^2t .\ tsin^2t + 3 cos^2t. t sintdt\\\\= 27 \int^{2\pi }_0 \dfrac{81}{4}sin^22t+ 27[-cos^3t]^{2\pi }_0\\\\= 27 \times \dfrac{81}{4} \int^{2\pi }_{0} (\dfrac{1-cos4t}{2})\\\\= \dfrac{2187}{8}[t-\dfrac{1}{4}sin4t]^{2\pi }_0\\\\= \dfrac{2187}{8}[2\pi -\dfrac{1}{4}sin4(2\pi )- 0-\dfrac{1}{4}sin4(0 )]\\\\= \dfrac{2187\pi }{4}[/tex]
Hence, By using Stokes Theorem the required S curl F.ds is [tex]\dfrac{2187\pi }{4}[/tex].
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