Final answer:
To find the area of the surface, calculate the surface integral ∬_S ds, where S is the surface and ds is the element of surface area. Calculate the unit normal vector n(u,v) as the cross product of the partial derivatives of r with respect to u and v. Then, integrate the element of surface area over the given range to obtain the area A = 16π.
Step-by-step explanation:
To find the area of the given surface, we need to calculate the surface integral. The surface integral is given by ∬_S ds, where S is the surface and ds is the element of surface area. In this case, the surface is defined by the equation r(u,v) = ⟨8cosu, 8sinu, v⟩, with 0 ≤ u ≤ 2π and 0 ≤ v ≤ 2.
First, we need to find the unit normal vector to the surface, which can be calculated as:
n(u,v) = r_u × r_v
where r_u is the partial derivative of r with respect to u and r_v is the partial derivative of r with respect to v. After calculating the partial derivatives, we can obtain the unit normal vector:
n(u,v) = ⟨-8sinu, 8cosu, 0⟩
Next, we calculate the magnitude of n(u,v) to get the element of surface area:
ds = |n(u,v)| dudv = 8 du dv
Finally, we integrate ds over the given range of u and v:
A = ∬_S ds = ∫_{0}^{2π} ∫_{0}^{2} 8 du dv
Simplifying the integral gives us:
A = 16π
Learn more about Surface Area