** 22.T
Centroids and moments of inertia II** <--(READ ME)

** Reference:** Stewart 13.4, 13.8, 8.4

** Keywords: ** Center of mass, centroid, moment of inertia,
double integrals, polar coordinates,
triple integrals, cylindrical coordinates, symmetry

** Learning Objectives:** The student will be able to:

- Define the center of mass of a thin lamina, and the centroid of a planar region
- Express as double integrals the mass of a thin lamina, its center of mass, and its moment of inertia with respect to a given axis
- Express a double integral as an iterated integral in polar coordinates
- Decide when polar coordinates are appropriate for evaluating a double integral
- Express a triple integral in cylindrical coordinates, and decide when cylindrical coordinates are appropriate for evaluating a triple integral
- Find the centroid and moments of inertia in simple cases, using polar and cylindrical coordinates

** 22.R Differential equations:
Direction fields, Euler's method**

** Reference:** Stewart 8.1, CalcLabs with Maple V Chapter 11 and page
185, Stewart 15.1

** Keywords: ** Ordinary differential equation (ODE),
solution to an ODE, direction field, analytical solution,
numerical solution, graphical solution, direction field, isocline,
Euler's method, numerical methods for ODEs

** Learning Objectives:** The student will be able to:

- Recognize a first order differential equation
- Verify by direct substitution that a function is a solution to a differential equation, or to an initial value problem
- Use Maple's dsolve command to find analytical and numerical solutions to differential equations
- Use Maple to produce a direction field for a first order differential equation
- Set up initial value problems for simple applications, such as those leading to exponential growth and decay
- Draw a direction field for a simple differential equation by plotting points or by using the method of isoclines
- Sketch the solution to an initial value problem using a direction field
- Use Euler's method
- Explain Euler's method graphically, using a direction field