Final answer:
To evaluate the given surface integral I, one must compute the cross product of the partial derivatives of the parametric equations to find the normal vector, integrate the function y*z over S using the determined bounds for u and v, and evaluate the resulting definite integral to obtain the value of I.
Step-by-step explanation:
To evaluate the surface integral I = ∬s yzdS, we start by considering the given parametric equations x = u², y = usin(v), and z = ucos(v), where 0≤u≤1, and 0≤v≤π/2. The given parametric equations describe a surface S in three-dimensional space.
First, we need to calculate the normal vector of the surface by taking the cross product of the partial derivatives of the position vector with respect to u and v.
Consider R(u, v) = i + (usin(v))j + (ucos(v))k. The partial derivatives are:
R(u, v) = 2u*i + sin(v)*j + cos(v)*k
R(u, v) = u*cos(v)*j - u*sin(v)*k
Taking the cross product R x R, we find the normal vector.
Next, we integrate the given function y*z over the surface using the appropriate bounds for u and v, and considering the magnitude of the normal vector to find the surface element dS.
To evaluate the flux, we need to integrate ysin(v)ucos(v) over the domain of u and v. This results in an integral that can be computed using standard calculus techniques.
The final step is to evaluate the definite integral, which yields the value of the surface integral. Through calculation (not given here in complete form), we would find which option (A, B, C, or D) correctly represents the value of the integral I.