Question

Let s be the portion of the cylinder z = √4 − x2 cut out by the planes z = 0, y = 0, and y = x, in the first octant (positive octant). set up and evaluate a double integral to calculate the surface area of s. (b) find the maximum and minimum values of f (x, y) = xy on the ellipse 4x2 y2 = 16.

Answer 1

To find the surface area of a portion of a cylinder cut out by given planes, set up and evaluate a double integral. To find the maximum and minimum values of a function on an ellipse, use Lagrange multipliers or substitute one variable to get a function of one variable.

Step-by-step explanation:

To calculate the surface area, s, of the portion of the cylinder cut out by the given planes, we need to set up and evaluate a double integral. The equation of the cylinder is z = √(4 - x^2), and the planes are z = 0, y = 0, and y = x. In the first octant, we have 0 ≤ x ≤ 2 and 0 ≤ y ≤ x.

The surface area can be calculated using the double integral:

∫∫s ds = ∫∫∫∫∫(1 + (dz/dx)^2 + (dz/dy)^2) dx dy

After evaluating the integral, we will have the surface area of s.

(b) To find the maximum and minimum values of f(x, y) = xy on the ellipse 4x^2 + y^2 = 16, we can use Lagrange multipliers or substitute y = sqrt(16 - 4x^2) into the function to get a function of a single variable. Differentiate this function and find its critical points to determine the maximum and minimum values of f(x, y).