Final answer:
To evaluate the surface integral (∬_S (x + y + z) , dA) for the given surface defined by the equation x^2 + y^2 = 9 and the range of z from 0 to 4, we need to find the surface area (dA) and integrate the given function over the surface. We first find the surface area vector and then evaluate the double integral by integrating over the y and z variables using the equation of the surface. The final answer can be obtained by evaluating the integral.
Step-by-step explanation:
To evaluate the surface integral (∬S (x + y + z) , dA), we need to find the surface area (dA) and integrate the given function (x + y + z) over the surface. The surface (S) is defined by the equation x^2 + y^2 = 9 and the range of z is 0 to 4. Let's first find the surface area (dA).
The surface (S) is a circular area with radius 3, lying in the xy-plane. Since the unit normal is pointing outwards, the surface area vector can be written as dA = -n_x dy dz. Using the equation of the surface, we can express y in terms of x and z as y = sqrt(9 - x^2). Therefore, the surface area vector becomes dA = -n_x(sqrt(9 - x^2))dydz.
Now, let's integrate the function (x + y + z) over the surface S. The integral becomes ∬S (x + y + z) , dA = ∫∫ (x + y + z)(-n_x(sqrt(9 - x^2))dydz).
Integrating over the y and z variables from 0 to 4, and using the equation y = sqrt(9 - x^2), we can evaluate the surface integral to get the final answer.