Let : DCR² → R³ be a nice parametrization of surface S. We introduce the following notation: (a) Show that E = ||T,||², F=T₁ Tv, G = ||T,||². ||Tux Tv||= √√EG - F². This gives an alternative formula for the surface area: A(S) = VEG-F2dS What happens to this formula if T and T₁, are orthogonal? (b) The sphere of radius R is parametrized by (u,v) = (Rsin(u) cos(v), R sin(u) sin(v), Rcos(u)), 0 ≤ u≤π, 0 ≤v≤2π. Use the formula above to find its surface area. (c) Determine the surface area of the portion of x+4y+8z = 4 that is inside the cylinder x² + y² = 16.

Let : DCR² → R³ be a nice parametrization of surface S. We introduce the following notation: (a) Show that E = ||T,||², F=T₁ Tv, G = ||T,||². ||Tux Tv||= √√EG - F². This gives an alternative formula for the surface area: A(S) = VEG-F2dS What happens to this formula if T and T₁, are orthogonal? (b) The sphere of radius R is parametrized by (u,v) = (Rsin(u) cos(v), R sin(u) sin(v), Rcos(u)), 0 ≤ u≤π, 0 ≤v≤2π. Use the formula above to find its surface area. (c) Determine the surface area of the portion of x+4y+8z = 4 that is inside the cylinder x² + y² = 16.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 18T

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