Use the free DPGraph viewer to see the figures.
Solids of integration from Section 6.2:
problem 28 and
problem 34, and the
I suggest that you click on the "Animate" menu in DPGraph. Look in particular at option number 2 under animate, where it tells how to slice throught the figure. This should help in setting up your integrals.
Here are some figures I would have liked to show to the class for section 9.1:
You can look under "Edit" to see the default values of the parameters a, b, c, and d.
You cannot change them in edit, unless you have the full-featured dpgraph (and not the
free DPGraph viewer). However, you can change the parameters with the scrollbar.
Point and box (textbook view). You can use the Scrollbar to move the position of the point (a,b,c).
Point and box (standard view). You can use the Scrollbar to move the position of the point (a,b,c).
Sphere. You can use the Scrollbar to move the position of the center of the sphere (a,b,c), and the radius d.
The following are fun graphs that I made (also see the
DPGraph gallery for many interesting graphs):
A movie of a drumhead vibrating in its fundamental mode.
A movie of a drumhead vibrating in a different mode.
Do you remember the twilight zone?
This is the
from problem 19.
(To be honest, this figure won't help you do the problem. You don't need to visualize the figure
to do the problem.)
Here is another parallelepiped. It looks better on the computer screen, even at a lower resolution, because the numbers in the defining equations are smaller. It is called ``Spat'' by Rainer Wonisch. I found it at the DPGraph library
Here is the parallelepiped I made and brought to class. The vectors along the edges are a = <2, 0, 0>, b = <1, 2, 0>, and c = <1, 1, 1>.
Here are some figures for Section 9.6:
z = 3 - x - y. A plane. (I did this example in class, but didn't show the figure.)
z = f(x,y) = |y|. A generalized cylinder. (Side view with no perspective). I showed this example in class.
z = f(x,y) = |y|. A generalized cylinder. (Textbook viewpoint with perspective).
z = |y| - |x|. I showed this in class.
z = y^2 - x^2. A hyperbolic paraboloid. (I didn't get to this in class. It is example 7 in the book.)
The following figures are not graphs of functions, because we cannot solve for a unique z = f(x,y). They are quadratic surfaces.
x^2 + z^2 = y^2. A cone. (I showed this in class.)
x^2 + z^2 = y^2 + 1. A hyperboloid of one sheet.
x^2 + z^2 = y^2 - 1. A hyperboloid of two sheets.
x^2 + z^2 = y^2 + a. Use scrollbar to change a.
x^2 + z^2 = y^2 + sin(time/2)^3. Movie showing hyperboloid of one and two sheets, and the kissing cones. (I showed this in class.)
Here are some figures for Section 10.1:
Figure for Problem 28.
Figure for Problem 29.
Figure for Problem 30.
Helices   Note: With the latest version of DPGraph these helices are much easier to draw!
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