Lab 01: Areas and Volumes

0. Objectives

In this lab we will use Mathematica to explore the concepts of areas between curves and volumes of rotation using integrals.  If you have not read the appropriate section in your book, please spend a few minutes to do so now.

1. In the beginning

First let’s look at an example.  Suppose we want to find the area between the graphs of the functions y=2x+1 and y=x^2-x+1.  A good place to start would be to graph the functions to get an idea of what we’re dealing with. Enter the two function into Mathematica and graph them on the interval [-5,5].

f[x_]=2*x+1;
g[x_]=x^2-x+1;
Plot[{f[x],g[x]},{x,-5,5},PlotRange->10,PlotStyle->{Blue,Red}]

Now use the Solve command to find the intersection points:

ip=Solve[f[x]==g[x],x]

You should get an answer of 0 and 3.  Then, based on the graph, the area between the curves is given by the following integral:

(∫_0^3)(f[x]-g[x])\[DifferentialD]x

We could also use the Integrate command:

Integrate[f[x]-g[x],{x,0,3}]

Now lets find the volume of rotation of this area rotated about the x-axis.  Using the washer method, this volume is given by:

V=π*(∫_0^3)(((f[x])^2)-((g[x])^2))\[DifferentialD]x

or by:

V=Pi*Integrate[f[x]^2-g[x]^2,{x,0,3}]

Now suppose that we wanted to rotate the area about the line x = 10.  In this case a representative slice, using vertical rectangles, would be a shell.  The radius of the shell would be r = 10 - x, the height of the shell would be f[x] - g[x], and the thickness would be dx, giving us a volume of:

V=2*π*(∫_0^3)(10-x)*(f[x]-g[x])\[DifferentialD]x

Notice that the limits of integration are all the same in the above area and volume problems because the region we’re working with doesn’t change and we are using vertical rectangles to model the incremental area and volumes.

2. Fun with Infinities

Next look at the function h[x] = (x^(-2/3)).  We can define the area under the curve on the interval [1,b] by:

h[x_]=x^(-2/3)
area[b_]=Integrate[h[x],{x,1,b},GenerateConditions->False]

Now plot this area function on the intervals [1,10], [1,100], and [1,1000].  Can you guess what the limit of the area function will be as b goes to infinity?  Look up the Limit command in the Documentation Center and use it to validate (or refute) your guess.

Next define the volume of rotation about the x-axis for this function h, on the interval [1,b].  Try plotting this volume function on the same intervals as above and see what the limiting volume is for this function.

Tasks

Open a new notebook using the File tab and clear all previous definitions by Quitting the local Kernel under the Evaluation tab.  Then complete the following tasks in the new notebook, print your results and turn the printed work in for credit.  Your finished product should include comments using the text style.

Use Mathematica to graph the following regions bounded by the given curves (you may want to use the ContourPlot command instead of the Plot command).  Decide whether to integrate with respect to x or y.  Then find the area of the region.  After printing, by hand shade the region and sketch a representative rectangle for each region.

y=(x^2)-2x, y=x+4.

4x+(y^2)=12, y=x.

y=(x/((((x^2)+1))^2)), y=(x^5)-x, x≥0.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.  Graph the region and after printing sketch by hand a typical disk, washer or shell.

y=(e^x), y=1, x=2,  about the x-axis.

y=(e^x), y=1, x=2,  about the y-axis.

x=(y^2), x+y=2, about the line x=-1.