Find the area of the surface: The part of the plane

Question

x + 2y + 3z = 1 that lies inside the cylinder x^2 + y^2 = 7?

Answer 1

Plane equation x + 2y + 3z = 1

x² + y² = 7

r² = 7

The cross section of a cylinder is a circle of radius √7, which has area 7π.

The vector < 1, 2, 3 > is perpendicular to the plane, so the angle the plane makes with the xy - plane is the same  as the angle θ that < 1, 2, 3 > makes with < 0, 0, 1 >

< 1, 2, 3 > · < 0, 0, 1 > = √14 cosθ

3 = √14 cosθ

cosθ = 3/√14

Thus required area is 7π/cosθ

= 7π/(3/√14)

= (7π√14)/3

Surface area is (7π√14)/3 .