## Week 22

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