# Math 141 - Week 7 Notes Monday, February 22, 1999

Topics for this week -

1. Lagrange Multiplier
2. Double integrals over rectangles
3. Iterated Integrals
4. Double integrals over general regions
5. Double integrals in polar coordinates

Topics for next week -

1. Double integrals in polar coordinates
2. Applications of double integrals
3. Surface area
4. Separable and homogeneous differential equations
5. First-order linear differential equations

Examples -

1. Find the maximum and minimum values of x2-2*y2 on the unit disk. Include checking the boundary!
2. Classify all critical points of (x2+2y2) exp(-(x2+y2)).
3. A pentagon is composed of a rectangle surmounted by an isosceles triangle (so parameters are the length/width of the rectangle and the base angle(s) of the triangle) -- given a fixed perimeter P, maximize the area.
4. Find shortest distance from (1,2,0) to the elliptic cone z2 = x 2 + 2y2
5. Minimize x4+y4+z4 on the plane x+y+z=1.
6. Evaluate the double integral of yex on the triangular region with vertices (0,0),(2,4),(6,0).
7. Evaluate int(int(y*cos(x^2),x=y^2..9),y=0..3);
8. Evaluate int(exp(-x^2),x=0..infinity);

Seventh Homework Assignment - due on FRIDAY, March 5.

1. Reading: In Chapter 12, read sections 12.7,12.8. In Chapter 13, read sections 13.1, 13.2, 13.3, 13.4.
2. Make sure you can do all of the Core problems for sections 12.7,12.8, 13.1, 13.2, 13.3 and 13.4. But write up only the following to be handed in ( from the old final exams):
• Section 12.7, p. 819, #8, 12, 26, 30, 42, 48 (it's ok to use Maple to help with the algebra!)
• Section 12.8, p. 826, # 8, 16, 20, 30, 40.
• Section 13.1, p.837, #5,
• Section 13.2, p.842, #6,10,18,28,33
• Section 13.3, p.850, #5,8,12,16,22,28,36,40,45

The idea behind Lagrange Multiplier