Calculus 3 Parametric

Question

Find the surface area of the part of the plane  that lies inside the cylinder 

Answer 1

A = ∫∫_Ω [ 1 + ( ∂z/∂x)² + ( ∂z/∂y)² ] ^1/2 d A

z = 3 - 3x/2 -y/2 ⇒ ( ∂z/∂x)² = 4 ( ∂z/∂y)² = 9

A = ∫∫_Ω [ 1 4 ] ^1/2 d A =

= ∫₀^2π ∫₀⁴ √14 r d r d θ =

= √14/2 ∫₀^2π [ r²] | ₀⁴ d θ =

= √14· 8 ·2·π =16 √14 π

Answer 2

This will be an ellipse. The minor axis has length equal to twice the radius of the circle, 2*4 = 8

The major axis can be easily determined. Ignore the 6. 2x+3y is maximized when x:y = 2:3

Then, x = 8/√13, y = x = 12/√13, z = 52/√13, and the major axis has length 2 * √(x^2+y^2+z^2) =

2 * √(8/√13^2+12/√13^2+52/√13^2) = 8√14

Thus, the surface area = π * minor axis * major axis/4 = π * 8 * 8√14 / 4 = 64√14 π