# 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

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

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