6.1 More About Areas

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Sources and references for some of these problems include
    James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001
    Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998
    Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993
    Selwyn Hollis, "CalcLabs with
Mathematica" for Stewart's "Single Variable Calculus, Concepts and Contexts, Second Edition," Brooks/Cole, 2001

Area of a Region Bounded by Two Curves

Reference: In this example we follow the programming style used by Selwyn Hollis in "CalcLabs with Mathematica," Section 5.1, "Area."

Let's calculate the area of one of the enclosed regions bounded above by the Sin finction and below by the Cos function.

In[11]:=

<<Graphics`Colors`

In[12]:=

Clear[f, g, x]  f[x_] := Sin[x] g[x_] := Cos[x]  Plot[{f[x], g[x]}, {x, 0, 2π}, PlotStyle {LightCoral, LightCadmiumRed}, Background->Mint] ;

[Graphics:HTMLFiles/6.1_moreAreas_3.gif]

Mathematica has a "FilledPlot" function for displaying such regions.

In[16]:=

<<Graphics`FilledPlot`

In[17]:=

FilledPlot[{f[x], g[x]}, {x, 0, 2π}, Background->Mint] ;

[Graphics:HTMLFiles/6.1_moreAreas_6.gif]

A vertical cross-section through the point x has length f[x] - g[x], for π/4 ≤ x ≤ 5π/4.

In[18]:=

f[x] - g[x]

Out[18]=

-Cos[x] + Sin[x]

Therefore the enclosed area is given by

In[19]:=

∫_ (π/4)^(5π/4) (f[x] - g[x]) x %//N

Out[19]=

2 2^(1/2)

Out[20]=

2.82843


Created by Mathematica  (April 29, 2004)