|
Examples For Exam IV
| 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
7 free download. |
Maple
Worksheet on Partial Derivatives and Multiple Integrals
Calculus
II volume review examples
| Volume Example 1a
Find the volume of the region of space (or solid)
above the
graph of the function g(x,y) and below the graph of the function f (x,y)
with the bounds on x and y as given below. Here is a DPGraph
picture of the solid (or region of space).
Click here or
on the picture to see a Maple worksheet, including pictures, investigating
this example including approximations to the volume using a sum of the
volumes of approximating rectangular boxes. The region along with
one approximating rectangular box is pictured at the right. |
Click here
for an animation showing different views and the heights of each
approximating box. QT version
|
| Volume Example 1b
Click here or on the
picture on the right to see an animation showing the region being filled
with 10 approximating boxes. Quicktime
version Click here to see the
Maple worksheet for this example which includes constructing the pictures
for the animation. PowerPoint Show |
 |
| Volume Example 1c
Click here to see a Maple
worksheet for this example which includes the construction of pictures
used in this animation which shows the
volume approximated by 15 rectangular boxes. Quicktime
version The picture
at the right shows the region of the xy-coordinate plane partitioned into
15 equal sized rectangles with the height of each approximating rectangular
box shown in red. |
 |
| Section 13.2 #28 (6e 24) DPGraph
Picture
Find the volume of the solid in the
first octant bounded by the graphs of
The
volume would be computed by integrating z = 4 - y2 over the
region shown on the right. |
Click on the picture above to see a picture of the
solid. |
| Section 13.2 #32 (6e 28) DPGraph
Picture
Find the volume of the solid in the
first octant bounded by the graphs of
|
Click on the picture above to see a picture of the
solid. |
Area of a
Polar Region (t is theta in the animations)
| Section 13.3 (6e #10) animation
QT
Use a double integral to find the area of the
region bounded by the graph at the right, function given below.
|
Click on the picture to see another plot with
terminal sides of angles theta drawn in blue extending a distance r from the
origin (pole). Click here
to see the graph on "polar graph paper".
|
| Section 13.3 #42 (6e 12)
animation
QT
Use a double integral to find the area of the
region indicated on the right, function given below.
|
 Click here
to see the graph on "polar graph paper".
|
|
Area of a
Rose Petal
Find the area
of one petal of the graph of
Notice the
limits of integration.
Click on the picture at the right to enlarge. |
 |
| Section 13.3 #20 (6e 18)
|
Click here to see an animated point
moving along the curve. Theta is t in the animation and is taking on
values from zero to pi. QT
Click here
to see the graph on "polar graph paper".
The double integral would represent the volume of
the region between the two surfaces shown above. That would be the figure
indicated in this DPGraphPicture.
Here is a DPGraphPicture
showing the intersecting surfaces. Here is
a DPGraph picture of the surface
z = x2 graphed over the region of integration done using
cylindrical coordinates and here is the
same surface along with the
cylinder
x2
+ y2 = 4y graphed transparently making use of cylindrical
coordinates. The graphs have actually been shifted in order to be
able to draw them this way--that is why "Box" has been turned off.
Maple
Worksheet |
| Mass, Moments, and Center of Mass
of a Planar Lamina
|
DPGraph
picture of the plate with a colorization that somewhat corresponds to
the density function although colors do repeat on the right side of the
plate. Click on the plate above to see a colorization that perhaps
better represents the density function. Click
here for a density function colorization taken from a Maple contour
plot.
|
| Surface Area Example
Find the area of the surface defined by the
function f(x,y) over the region defined below. Click
here to see a brief discussion of the development of the surface area
integral formula used in this problem along with pictures (Powerpoint
Presentation). Click on the picture
at the right to see an enlargement. Click
here to see an animation showing a variety of views. Quicktime
version
|
Here is the Maple
worksheet that creates pictures that zoom in on the point of tangency
shown above. QT version
|
| The picture below on the left shows the
surface. The picture below on the right shows an approximation to
the surface using 15 "pieces of tangent planes". Click
here to see a Maple worksheet further investigating the surface area
being looked at in this problem including approximating the surface area
using 15, 60, 240, and 960 "pieces of tangent planes".. Here is an animation
going from the picture below on the left to the picture below on the
right. Quicktime Version. |
Click here
to see an animation showing different views of four approximating pieces
of planes. QT version
|
Click here
to see an animation showing different views of the fifteen approximating
pieces of planes. QT version
|
| Section 13.5 #37 (6e 35)
Surface Area Example
Find the surface area of the solid of
intersection of the cylinders whose equations are x2 + z2
= 1 and y2 + z2 = 1.
DPGraphPicture
of the intersecting cylinders
DPGraphPicture
of the surface
DPGraphPicture
of the portion of the surface above the region at the right
DPGraphPicture
of the surface and the plane tangent to the surface at the point
(0.6,0.2,0.8) over the region specified below.
Zoom-in-out
on the plane tangent to the surface x2 + z2 = 1 at
the point given below:
Use the scrollbar and activate a. If you change a.maximum to 100 you
will not be able to tell the difference between the surface and the
tangent plane when a = 100.
A picture of the triangular region over which the
double integral will be iterated is shown at the right above. |
DPGraph
view from the top of the portion of the surface over the region shown
above
|
|
|
Click the picture to enlarge.
Here is a Maple
Worksheet that graphs the cylinders and computes the desired surface
area.
|
| Surface Area--Extra Credit
DPGraph
Picture You can use the scrollbar to vary b from 0 to 16. |
 |
| Volume Example
Find the volume of the wall bounded by the graphs
of the surfaces given below. The region over which we would be
iterating the double integral is pictured at the right. Click on DPGraphPicture
to see a 3D representation of the wall. Click on DPGraphPicture2
to see the wall placed in an xyz-coordinate system.
|
Click on the picture above to see a Maple picture
of the wall.
|
| Volume Example
Find the volume of the region of
space bounded by the graphs of z = x2+2y2, z = 2, and z =
8. See the graph on the right. Click on the picture to see an
animation. DPGraphPicture
|
 |
|
Method
1
One option would be to integrate over the small
ellipse, the level curve for z=2, with z going from 2 to 8 (i.e., the area
of the small ellipse times 6) and add to this the result of integrating
over the region between the two ellipses (between the level curve for z=2
and the level curve for z=8 which I am calling region R) with z going from
x2 + y2 to 8.
|
| Method 2
An alternative would be to find the volume of the
whole paraboloid up to z=8 and then subtract from this the volume up to
z=2.
|
|
Method
3
Another option is to find the cross section area
of a slice perpendicular to the z-axis and integrate this cross section
area function of z from 2 to 8.
|
| Mass, Moments, Center Of Mass of a
Solid
DPGraphPicture1
DPGraphPicture2
|
Pictured
below is the intersection of the solid with the xy-coordinate plane.
Click on the picture to see a Maple picture of the solid.
Here is a DPGraph
transparent picture of the surfaces along with the center of mass.
Here is a DPGraph
picture of the solid with the color corresponding to the density
function. |
| Section 13.7 #13
|
The picture above is meant to represent 1/4 of
the region we are integrating over. Click
here or on the picture to see an enlargement. Click on DPpicture
to see a DPGraph visualization of the full region we are integrating over.
Below is a picture of the whole region we are
integrating over.
 |
| Section 13.7 #14
|
Click here or on the picture above to see an
enlargement of the diagram. Click on DPpicture
to see a 3D graphic of the region we are integrating over or click on DPGraphPicture
to see the various surfaces extended.
|
| Section 13.7 #15 with
a = 2 DPpicture
|
Click on the figure above to see an enlarged
version. The top part of the figure is representing 1/4 of the
region being integrated over (shown below)
 .
|
Section 13.7 #18 (6e 26)
DPpicture DPPicture2
Section 13.7 #32(Variation, a = 2, b = 4)
(6e 30) DPpicture DPpictureTransparent
| Another Volume Example--Extra
Credit
DPGraphPicture
DPGraphPic3Surfaces
The picture at the right shows the level curves
for the first two surfaces described above corresponding to z = 5,
cylinder in green,
paraboloid in red. |
 |
| Section 13.8 #19
|
|
Return
|
