Thermodynamics and Chemistry

APPENDIXE CALCULUSREVIEW 480

E.3 Integrals

Letf be a function of the variablex. Imagine the range ofxbetween the limitsx^0 andx^00
to be divided into many small increments of sizeÅxi.iD1; 2; : : :/. Letfibe the value of
fwhenxis in the middle of the range of theith increment. Then theintegral
Zx 00

x^0

fdx

is the sum

P

ifiÅxiin the limit as eachÅxiapproaches zero and the number of terms in
the sum approaches infinity. The integral is also the area under a curve off plotted as a
function ofx, measured fromxDx^0 toxDx^00. The functionf is theintegrand, which
is integrated over the integration variablex.
This book uses the following integrals:
Zx 00

x^0

dxDx^00 x^0

Zx 00

x^0

dx

x

Dln

x^00

x^0

Zx 00

x^0

xadxD

1

aC 1



.x^00 /aC^1 .x^0 /aC^1



(ais a constant other than 1 )

Zx 00

x^0

dx

axCb

D

1

a

ln

ax^00 Cb

ax^0 Cb

(ais a constant)

Here are examples of the use of the expression for the third integral withaset equal to 1
and to 2 :
Zx 00

x^0

xdxD

1

2



.x^00 /^2 .x^0 /^2



Zx 00

x^0

dx

x^2

D



1

x^00

1

x^0



E.4 Line Integrals

Aline integralis an integral with an implicit single integration variable that constraints the
integration to a path.
The most frequently-seen line integral in this book,

R

pdV, will serve as an example.
The integral can be evaluated in three different ways:
1.The integrandpcan be expressed as a function of the integration variableV, so that
there is only one variable. For example, ifpequalsc=Vwherecis a constant, the
line integral is given by

R

pdV Dc

RV 2

V 1 .1=V /dV Dcln.V^2 =V^1 /.

2.IfpandV can be written as functions of another variable, such as time, that coordi-

nates their values so that they follow the desired path, this new variable becomes the

integration variable.

3.The desired path can be drawn as a curve on a plot ofpversusV; then

R

pdV is

equal in value to the area under the curve.