By applying Stokes theorem, the value of the integral is equal to -52π.
How to use the Stokes theorem?
First of all, we would use Stokes theorem to convert the integral as follows: int s (Delta*F) .mds= int c Fdr
where:
c represents the boundary circle on the xy-plane.
Note: The bounding curve (x ^ 2 + y ^ 2 = 49) is a circle located at the origin with a radius of 7.
Thus, the parametric equation for the circle is given by:
r(t)=7 cos t hat i + 7sin tj t is from 0 to 2Pi
Differentiating r(t) with respect to t, we have: r^ prime (t)=-7 sin t hat i + 7cos tj .
Next, we would compute F(r(t)) :
F(r(t))=8(7 sint) hat i + 5(7cos t) * j z = 0 delta*z = 0'
By taking the dot product, we have: F(r(t)) * r' * (t) = - 392sin^2 t + 245cos^2 t
Note: The double-angle formula for cos is cos(2t) = 2cos^2 t - 1
Now, we can integrate from 0 to 2π: integrate (- 392sin^2 t + 245cos^2 t) dt from 0 to 2pi
integrate (- 392/7 * (1 - cos 2t) + 245/7 * (1 + cos 2t)) dt from 0 to 2pi
-392 7 int 0 ^ 2 pi (1 - cos 2t) + 245/7 * integrate (1 + cos 2t) dt from 0 to 2pi
int s ( Delta* F). m * (ds = - 56[2pi + 0] + 35[2pi + 0])
int s ( Delta* F). m * (ds = - 112pi + 70pi)
int s ( Delta* F). m * ds = - 52pi iint m( nabla* f)* ds = - 52pi .