Question Sheet 7 - Challenge Problem Example Answers

Often in physics, we will want to integrate over the surface or volume of a sphere. Determining the limits for this in Cartesian coordinates can be time consuming. Mathematica has a built-in way of specify a surface or volume easily. Google ‘mathematica 10 integrate over regions’ to find the wolfram help file that shows you how to do this.This should lead you to a webpage on the Wolfram Language Documentation Centre which is a very helpful resource. Often it can be easier to google something than search it in the helpfile.]

Graphics3D[{Blue,Opacity[1.0],Ball[{0,0,0},0.15]},Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-0.3,0.3},{-0.3,0.3},{-0.3,0.3}},Boxed->False,AxesStyle->{{Thick,Black},{Thick,Black},{Thick,Black}}]

1a) What is the surface area of a sphere?

4((πr)^2)

b) What is the surface area of a sphere of radius 2? What is the volume of a sphere of radius 2?

Surface area = 16π (cm^2)
Volume = (4/3)π (r^3)= (32/3)π (cm^3)

c) What is the difference between Sphere[ ] and Ball[ ] in Mathematica?

A sphere only includes the surface, whereas a ball includes the entire volume.

d) Calculate the surface area and volume using the integration command. Do your answers match your prediction?

Integrate[1,{x,y,z}∈Sphere[{0,0,0},2]]

Integrate[1,{x,y,z}∈Ball[{0,0,0},2]]