Surface Integrals – On a closed domain – Exercise 4782

Exercise

Let E be the bounded area between the XY plane and the surface

$z=4-x^2-y^2$

Let S be the surface area of E.

Calculate the integral

$\int\int_S F\cdot\hat{n} ds$

Or in another notation

$\int\int_S F\cdot N dA$

Where the vector field F is

$F=(xz\sin(yz)+x^3,\cos(yz),3zy^2-e^{x^2+y^2})$

Final Answer

$\int\int_S F\cdot\hat{n} ds=32\pi$

Solution

Coming soon…

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