Respuesta :
The value of the surface integral ∫sf⋅ ds∫sf⋅ ds is (1-e)/4 if f=−4xyi x2j−4yzk f=−4xyi x2j−4yzk
What is integration?
It is defined as the mathematical calculation by which we can sum up all the smaller parts into a unit.
The surface integral is given by:
[tex]= \rm \int\limits \int_S\ {F.} \, ds[/tex]
F = −4xyi + x²j − 4yzk
P(x,y,z) = -4xy, Q(x,y,z) = x², R(x,y,z) = -4yz,
[tex]\rm g(x, y) = z=xe^y[/tex]
∂g/∂x = [tex]\rm \rm e^y[/tex]
∂g/∂y = [tex]\rm xe^y[/tex]
[tex]\rm \int\limits \int_S\ {F.} \, ds = \int\limits^1_0 \int\limits^1_0 [ {-P\frac{\partial g}{\partial x}} \, -Q\frac{\partial g}{\partial y} + R] dA[/tex]
Plug all values and solve we will get:
[tex]\rm \int\limits \int_S\ {F.} \, ds = \int _0^1\int _0^1\:\left(4xy\left(e^y\right)-x^2\left(xe^y\right)-4y\left(xe^y\right)\right)dydx[/tex]
[tex]\rm \int\limits \int_S\ {F.} \, ds = \dfrac{1-e}{4}[/tex]
Thus, the value of the surface integral ∫sf⋅ ds∫sf⋅ ds is (1-e)/4 if f=−4xyi x2j−4yzk f=−4xyi x2j−4yzk
Learn more about integration here:
brainly.com/question/18125359
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