Calculus 3 Timed Quiz 9

    Planar Lamina. A planar lamina is a flat plate that occupies a region in the plane that is so thin it can be regarded as _____  ____________. If is the lamina's _________ and is the _________ of  the region then is the __________ of the lamina (in units of mass per unit area). A lamina is __________ if its density is __________ over and nonhomogenuous if varies from __________ to __________.
    If is a continuous density function on the lamina corresponding to a plane region then the (total) mass of a lamina is given by Physicists also consider other types of density that can be treated in the same manner. For example, if an electric charge is distributed over a region and the charge __________ (in units of charge per unit area) is given by at a point in then the __________ charge is given by  

Example (Planar Lamina) Charge is distributed over the triangular region described by and so that the charge density at is measured in coulombs per square meter. Find the total charge.
    
    Solution.

    Moments and Center of Mass. The moment of an object about an axis measures the __________ of the object to __________ about that __________. It is defined as the product of the object's mass and the signed distance from the axis. The center of mass of the lamina covering a region is the point where the __________ can be concentrated without affecting the __________ and ; that is,
    The center of mass may also be thought of as the point from which the lamina may be __________ without movement. If is a __________  __________ function on a lamina corresponding to a plane region then the moments of mass with respect to the -axis and -axis, respectively, are



Furthermore, if is the mass of the lamina, the center of mass is where If the density is constant, the point is called the __________ of the region.

Example (Moments and Center of Mass of a Planar Lamina) Find the mass and the center of mass of a triangular lamina with vertices and if the density function is
    
    Solution.     

    Joint Probability Density Function. If and are both ______________ _________________ _____________, then the joint probability density function for two random variables and is a function of two variables such that for all and



where denotes the probability that is in the region Note that



Geometrically, may be thought of as the __________ under the surface above the region .

Example (Probability Density Functions) Suppose the joint probability density function for the random variable and is is modeled by



Find the probability that
    
    Solution.    

Example (Probability Density Functions) Suppose measures the time (in minutes) that a person stands in line at a certain bank and the duration (in minutes) of a routine transaction at the teller's window. You arrive at the bank to deposit a check. The joint probability density function for and is modeled by



Find the probability that you will complete your business at the bank within 8 minutes.
    
    Solution.

• print this page
• pages linking here
• related pages

Cite this as:
Calculus 3 Timed Quiz 9
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/calculus-3-timed-quiz-9.html