cclogo.gif (38585 bytes)   Seminole State Homepage   |   Calculus I   |   Calculus II    |    Calculus III   |   Diff Eq    |    Calculus Homepage cclogo.gif (38585 bytes)

 

CALCULUS III    EXAM IV NOTES AND LINKS

Click here to view practice problems for Exam IV.

 
         

 
Volume Examples Polar Coordinate Examples Mass, Moments, Center of Mass
Surface Area Examples Cylindrical, Spherical Coordinates Example Using The Jacobian
If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page.   QuickTime free download.

 
bullet Interesting Life and Death Applet
bullet Here are some terrific class notes for Calculus III from Paul Dawkins at Lamar University.
bullet Traces Animes  A really nice set of 2D and 3D WIMS graphing tools by XIAO Gang which will produce pictures and graphs that can then be saved and printed, viewed using other software such as Quicktime, or used on a web site.
bullet Three D Graphing Utility and other math resources including an Excel three D grapher.
bullet A visually appealing 3D Graphing Utility but you are limited in the types of graphs you can create.
bullet Allegany College of Maryland has a Calculus III course online.  Unit 4 is on Multiple Integration.  Click on Unit 4 at the top of the page when you get there. 
bullet Fabulous Mini-Putt Golf  Thanks to Chuck Logan for this one.
bullet DPGraph Animations:  Fire and Light, Fire and Light from the Top
bullet Fire and Light from the Top  (smaller step size)
bullet Examples
bullet Lake Tahoe CC Examples
bullet Maple Worksheet on Partial Derivatives and Multiple Integrals
bullet YouTube proof that the sum of all natural numbers is -1/12
bullet

Here is a DPGraph animation of a surface deforming into a variety of quadric surfaces.  You will need to download the free DPGraph Viewer to view it.  In this DPGraph picture of Quadrics you can use the scrollbar to activate "a" .  By using the scrollbar to vary the value of "a" you can see the various quadric surfaces shown in the animation.  The equation being graphed is given below.  

Here are some DPGraph pictures of a cylinder, a surface of revolution, and Quadric surfaces:  Cylinder, Surface of Revolution, Surface of Revolution 2, Ellipsoid, Hyperboloid(both), Elliptic Cone, Paraboloid (both).  You can click on the pictures below to see animations.  In this Maple Worksheet you can see the constructions of a variety of surfaces in space.   Here is a link to some surfaces of revolution on my Calculus II site along with volume and surface area computations.  Here is another Maple Worksheet relating to a surface of revolution and two potential generating curves.

14.1  You will need to be able to model the area of a region bounded by the graphs of given equations in terms of an iterated integral (that will satisfy the criteria for a double integral) and evaluate it.

14.2  Review Material:  You should also be able to compute the volume of a solid whose cross section area function is known (See Page 461, 426 in 7th ed, and these Quicktime movies by Bruce Simmons).  Another example of this would be a solid whose bottom is bounded by a circle and whose cross sections are squares.  Here is a java applet illustrating such a problem.  Calculus II volume review examples

 
14.2  You will need to be able to set up a double integral representing the volume of a solid (or region of space) bounded by the graphs of given equations, iterate it, and evaluate it.  Here is a graphic demonstrating the volume of the intersection of two cylinders.  Here is a nice presentation of Riemann Sum for a Double Integral and another one with animations submitted by Tony Nelson.  Here are two animations to introduce the concept of the Riemann Sum as applied to double integrals by Frank Wattenberg of Montana State University (click here) and T.J. Murphy of Oklahoma University (click here).  Click on my picture on the right to see an animation.  Here is another of my animations demonstrating the insertion of 10 approximating rectangular boxes to approximate the volume of a solid.  Quicktime version  
Click here to see a Maple worksheet relating to the animation above which also identifies the function involved, the region of the xy-coordinate plane to be integrated over, and approximations to the volume (along with the Maple code for the computations)  involving 10, 40, 160, and 640 approximating boxes.

 
14.3  You will need to be able to convert a double integral from rectangular coordinates to polar coordinates.  You will need to be able to use polar coordinates in computing areas and volumes.  Here is an ExploreMath activity to help you review polar coordinates.  The total area enclosed by the rose petals at the right is computed below.

 

14.4  You will need to be able to find the mass, moments of mass about the x- and y-axis, and center of mass of a planar lamina with a continuous density function.  Here is a cute little ExploreScience activity dealing with point masses on a planar surface.

14.5  You will need to be able to model the double integral needed to find the area of a given surface over a region of the xy-plane and in many cases to evaluate it either exactly or approximately.  Click here to zoom in on a point of tangency of a tangent plane (since we are using areas of pieces of tangent planes to develop the surface area integral) and here for the Quicktime version.  Click here to see again my DPGraph tangent plane zoom (closer zoom).  Here again is my DPGraph scrollbar controlled tangent plane zoom:  DPGraph of a sphere and a tangent plane--Use the scrollbar and activate "a" to zoom in on the point of tangency (closer zoom).  This DPGraph Picture of a surface over a region of the xy-coordinate plane shows the region of the xy-coordinate plane partitioned and the corresponding partitions of the surface.

14.6  You will need to be able to set up a triple integral representing the volume of a solid (or region of space) bounded by the graphs of given equations, iterate it, and evaluate it.  Here is a triple integral site which includes links to other triple integral examples and tutorials (submitted by Tony Nelson).  You will need to be able to find the mass, moments of mass about the xy-, yz-, and xz-planes, and center of mass of a solid with a continuous density function.  Here is another tutorial on multiple integration.  

14.7  This applet might help to introduce you to spherical coordinates.  You will need to be able to model triple integrals in cylindrical and spherical coordinates, iterate them, convert from rectangular to cylindrical and/or spherical coordinates, and compute volumes and masses using cylindrical and/or spherical coordinates.  Here are some java applets for graphing surfaces in rectangular, cylindrical, and spherical coordinates.

14.8  You will need to be able to use the Jacobian to implement a change of variables in evaluating an iterated double integral or in setting up and evaluating an iterated double integral with a volume interpretation as in exercises 17-22 on page 1047 (p1000 in 7th ed).  Here is another link from Tony Nelson.  This is an applet demonstrating visually the Jacobian.

Top of the page

 

 

 

 

 

          This site contains links to other Internet sites.  These links are not endorsements of any products or services in such sites, and no information  

           in such site has been endorsed or approved by this site.

        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats