Some observations

rlopez — Wed, 14 Nov 2012 20:33:09 Z

In 1(a), the value of the integral is zero because sin(Pi) is a factor in the integrand. I'm sure this is a typo, but if you take the integrand as just 1, you can get a sketch of the region of integration with the task template:

Calculus-Multivariate/Integration/Visualizing Regions of Integration/Cartesian 2-D
The help page for this template can be reached by executing the command: ?IntPlotCartesian2D

1(b) Unfortunately, I know of no algorithm by means of which the order of integration in a multiple integral can be changed. I always taught my students to draw the region, and from this sketch rebuild the integral with the alternate order.

2. The centroid is the center of mass when the mass is uniform. For this, try the task template:

Calculus-Multivariate/Integration/Center of Mass/Polar

The help page for this template can be reached by executing the command: ?CenterOfMassPlanarPolar

3. The following commands will set up and evaluate the integral in cylindrical coordinates:

q:=Int(r,[z=0..-r*sin(t),r=0..3,t=Pi..2*Pi]);

value(q);

The following task template was useful for drawing the region corresponding to the integral:

Calculus-Multivariate/Integration/Visualizing Regions of Integration/Cylindrical

The help page for this template can be reached by executing the command: ?IntPlotCylindrical

4. The following commands will set up and evaluate the integral in spherical coordinates:

q:=Int(rho^2*sin(phi),[rho=1/cos(phi)..2/cos(phi),phi=0..Pi/4,theta=0..2*Pi]);

value(q);

The following task template was useful for drawing the region corresponding to the integral:

Calculus-Multivariate/Integration/Visualizing Regions of Integration/Spherical

The help page for this templage can be reached by executing the command: ?IntPlotSpherical

The Table of Contents for the Task Templates is obtained by choosing Tasks/Browse from the Tools menu.

RJL Maplesoft

MaplePrimes - answers and comments on Question, I am in desperate need of assisstance in solving these problems. Help?

Some observations