Calculus III: Exam IV Notes and Links

Click here Links to an external site. to view practice problems for Exam IV.

 

         

 

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

 

bullet Interesting Life and Death Applet Links to an external site.
bullet Here are some terrific class notes for Calculus III Links to an external site. from Paul Dawkins at Lamar University.
bullet Traces Animes Links to an external site.  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 Links to an external site. and other math resources including an Excel three D grapher.
bullet A visually appealing 3D Graphing Utility Links to an external site. 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 Links to an external site..  Click on Unit 4 at the top of the page when you get there. 
bullet Fabulous Mini-Putt Golf Links to an external site.  Thanks to Chuck Logan for this one.
bullet DPGraph Animations:  Fire and Light Links to an external site., Fire and Light from the Top Links to an external site.
bullet Fire and Light from the Top Links to an external site.  (smaller step size)
bullet Examples Links to an external site.
bullet Lake Tahoe CC Examples Links to an external site.
bullet Maple Worksheet on Partial Derivatives and Multiple Integrals Links to an external site.
bullet YouTube proof Links to an external site. that the sum of all natural numbers is -1/12
bullet

Here is a DPGraph animation Links to an external site. of a surface deforming into a variety of quadric surfaces.  You will need to download the free DPGraph Viewer Links to an external site. to view it.  In this DPGraph picture of Quadrics Links to an external site. 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 Links to an external site., Surface of Revolution Links to an external site., Surface of Revolution 2 Links to an external site., Ellipsoid Links to an external site., Hyperboloid(both) Links to an external site., Elliptic Cone Links to an external site., Paraboloid (both) Links to an external site..  You can click on the pictures below to see animations.  In this Maple Worksheet Links to an external site. you can see the constructions of a variety of surfaces in space.   Here is a link to some surfaces of revolution Links to an external site. on my Calculus II site along with volume and surface area computations.  Here is another Maple Worksheet Links to an external site. 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 Links to an external site.).  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 Links to an external site..  Calculus II volume review examples Links to an external site.

 

14.2  You will need to be able to set up a double integral representing thevolume of a solid Links to an external site. (or region of space) bounded by the graphs of given equations, iterate it, and evaluate it.  Here is a graphic demonstrating thevolume of the intersection of two cylinders Links to an external site..  Here is a nice presentation of Riemann Sum for a Double Integral Links to an external site. and another one with animations Links to an external site.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 Links to an external site.) and T.J. Murphy of Oklahoma University (click here Links to an external site.).  Click on my picture on the right to see an animation.  Here is another Links to an external site. of my animations demonstrating the insertion of 10 approximating rectangular boxes to approximate the volume of a solid.  Quicktime version Links to an external site.  
Click here to see a Maple worksheet Links to an external site. 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 Links to an external site. 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 Links to an external site. 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 Links to an external site. of the xy-plane and in many cases to evaluate it either exactly or approximately.  Click here Links to an external site. 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 Links to an external site..  Click here Links to an external site. to see again my DPGraph tangent plane zoom (closer zoom Links to an external site.).  Here again is my DPGraph scrollbar controlled tangent plane zoom:  DPGraph of a sphere and a tangent plane Links to an external site.--Use the scrollbar and activate "a" to zoom in on the point of tangency (closer zoom Links to an external site.).  This DPGraph Picture Links to an external site. 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 Links to an external 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 Links to an external site..  

14.7  This applet might help to introduce you to spherical coordinates Links to an external site..  You will need to be able to model triple integrals in cylindrical Links to an external site. and spherical Links to an external site. 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 Links to an external site., cylindrical Links to an external site., and spherical Links to an external site. 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 Links to an external site..