University:
University of California, Berkeley
Course:
MATH 56 Linear Algebra
Academic year:

2024
Views:

202

Pages:

1
Author:

Clint W.

Definition: Suppose S is a surface w ith equation z = f (x, y ), where f h as continuous partial derivatives. The surface area of S is Pm Pn A(S) = limm,n→∞ i=1 j=1 △ Tij . Formula: The area of the surface with equation z = f (x, y), (x, y) ∈ D, where fx and fy are continuous, is RR p A(S) = [fx (x, y)]2 + [fy (x, y)]2 + 1 dA. D Formula with Alternative Notation RR q ∂z 2 ∂z 2 A(S) = 1 + ( ∂x ) + ( ∂y ) dA. D Exercise 1. Let R be the triangular region in the xy-plane with vertices (0,0,0), (0,1,0), and (1,1,0). Find the surface area of that portion of the graph z = 3x + y 2 that lies over R. (Swok Sec 17.4 Ex 1) Exercise 2. Find the surface area of the paraboloid z = 4 − x2 − y 2 for z ≥ 0. (Swok Sec 17.4 Ex 2) Class Exercise 1. Find the area of the surface. (#2,4,8,10,12) (a) The part of the plane 2x + 5y + z = 10 that lies inside the cylinder x2 + y 2 = 9 (b) The part of the surface z = 1 + 3x + 2y 2 that lies above the triangle with vertices (0,0), (0,1), and (2,1) (c) The surface z = 32 (x3/2 + y 3/2 ), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (d) The part of the sphere x2 + y 2 + z 2 = 4 that lies above the plane z = 1 (e) The part of the sphere x2 + y 2 + z 2 = 4z that lies inside the paraboloid z = x2 + y 2