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SoCoToAh
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First, this is my first time actually posting anything so hi PF!Second, I have been working out of Div, Grad, Curl and all that. This problem has me stumped for some reason. My answer never comes out to be the same as the books. If you could help me figure out where I am going wrong I would greatly appreciate it.
Evaluate the surface integral using the following equation.
\int\int\vec{F}(x,y,z)\hat{n}=\int\int-F_{x}∂S/∂x-F_{y}∂S/∂y+F_{z}dS
Where,
S(x,y,f(x,y))=z=1-x^{2}-y^{2}
above the xy plane
and
\vec{F}(x,y,z)=\hat{j}y+\hat{k}
After setting up the integral I attempt to switch to polar:
\int\int-F_{x}∂S/∂x-F_{y}∂S/∂y+F_{z}dS=\int\int2y^{2}+1 dxdy
=\int\int2((rSin[θ])^{2}+1)rdrdθ
from r=0 to r=1 and θ=0 to θ=2\pi
=\intSin^{2}[θ]/2dθ+\pi
\intSin^{2}[θ]dθ=\int(1-Cos[2θ])/2dθ
=θ/2-Sin[2θ]/4
∴
\intSin^{2}[θ]/2dθ+\pi=\pi/4+\pi
The book says the answer is \pi/2. Looking at what I have I I could say 1=r^{2}
Then \int\int2((rSin[θ])^{2}+1)rdrdθ=\int\int2((rSin[θ])^{2}+r^{2})rdrdθ and thus equal to \pi/2.
Where am I going wrong? Thanks in advance.
Homework Statement
Evaluate the surface integral using the following equation.
Homework Equations
\int\int\vec{F}(x,y,z)\hat{n}=\int\int-F_{x}∂S/∂x-F_{y}∂S/∂y+F_{z}dS
Where,
S(x,y,f(x,y))=z=1-x^{2}-y^{2}
above the xy plane
and
\vec{F}(x,y,z)=\hat{j}y+\hat{k}
The Attempt at a Solution
After setting up the integral I attempt to switch to polar:
\int\int-F_{x}∂S/∂x-F_{y}∂S/∂y+F_{z}dS=\int\int2y^{2}+1 dxdy
=\int\int2((rSin[θ])^{2}+1)rdrdθ
from r=0 to r=1 and θ=0 to θ=2\pi
=\intSin^{2}[θ]/2dθ+\pi
\intSin^{2}[θ]dθ=\int(1-Cos[2θ])/2dθ
=θ/2-Sin[2θ]/4
∴
\intSin^{2}[θ]/2dθ+\pi=\pi/4+\pi
The book says the answer is \pi/2. Looking at what I have I I could say 1=r^{2}
Then \int\int2((rSin[θ])^{2}+1)rdrdθ=\int\int2((rSin[θ])^{2}+r^{2})rdrdθ and thus equal to \pi/2.
Where am I going wrong? Thanks in advance.