Find the surface area of that part of the plane 10x+8y+z=4 that lies inside the elliptic cylinder x^2/100+y^2/81=1

Answer 1

z = 4 - 10x - 8y; z_x = - 10; z_y = - 8.

Surface area s = ∫∫_σdA = ∫∫_D√ + 100 + 64dxdy = √165∫∫_Ddxdy

x = 10rcosθ; y = 9rsinθ; 0 ≤ θ ≤ 2π; dxdy = 90rdrdθ

s = 90√165∫₀^2πdθ∫₀¹rdr = 90√165 = 90√165·2π·1/2 = 90π√165