# MATH 16B Lecture Notes - Lecture 8: Minimax, Multiple Integral

30 views1 pages
School
Department
Course Three-variable equations
Max/min of f(x, y, z) such that g(x, y, z) = 0
F(x, y, z, lambda) = f(x, y, z) + (lambda)g(x, y, z)
Calculate PD(x), PD(y), PD(z), PD(lambda)
Solve all equations when equal to 0
(a, b, c, d) solution --> (a, b, c) potential max/min
Example: Determine the maximum possible volume of a cuboid such that surface area is 6.
Volume = xyz; surface area = 2xy + 2xz + 2yz = 6 (assume max volume exists and x, y, z > 0)
Xyz = f(x, y, z); 2xy + 2xz + 2yz = 6 --> xy + xz + yz -3 = 0 = g(x, y, z)
F(x, y, z, lambda) = xyz + lambda(xy) + lambda(xz) + lambda (yz) - lambda(3)
PD(x) = yz + lambda(y) + lambda(z), PD(y) = xz + lambda(x) + lambda(z), PD(z) =
xy + lambda(x) + lambda(y), PD(lambda) = xy + xz + yz - 3
Yz + lambda(y) + lambda(z) = 0, xz + lambda(x) + lambda(z) = 0, xy + lambda(x) +
lambda(y) = 0, xy + xz + yz - 3 = 0
Lambda = -yz/(y+z), lambda = -xz/(x + z), lambda = -xy/(x + y) -->
-yz/(y + z) = -xz/(x + z) and -xz/(x + z) = -xy/(x + y)
-y/(y + z) = -x/(x + z) and -z/(x + z) = -y/(x + y) --> -y(x + z) = -x(y + z) and -z(x + y) = -y(x + z) --> -yz
= -xz and -xz = -xy -->
-yz = -xz and -xz = -xy --> y = x and y = z
Xy + xz + yz - 3 = 0 --> x^2 + x^2 + x^2 - 3 = 0 --> 3x^2 - 3 = 0 --> x^2 = 1 --> x = 1
Y = 1 and z = 1 and lambda = -1/2
(1, 1, 1, -1/2) is the only solution (x, y, z > 0)
Max must be at (1, 1, 1) so the max volume is 1
Double Integrals
f(x): single-variable function, continuous on [a, b]
Definite integral of f(x) = area above x-axis bounded by y = f(x) between a and b minus
area below x-axis bounded by y = f(x) between a and b
Fundamental Theorem: definite integral of f(x) = F(b) - F(a)
F'(x) = f(x)
Aim: Generalize to 2-variable functions
Replace [a, b] with rectangle R in xy-plane
Replace f(x) with f(x, y) continuous 2-variable function on R
Definition: Double integral of f(x, y) on R = volume above xy-plane bounded by z
= f(x, y) and R minus volume below xy-plane bounded by z = f(x, y) and R
How do we calculate the double integral of f(x, y) on R?
find more resources at oneclass.com
find more resources at oneclass.com
Unlock document

This preview shows half of the first page of the document.
Unlock all 1 pages and 3 million more documents.