Preface

Here are my online notes for my Calculus I course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus I or needing a refresher in some of the early topics in calculus. Ive tried to make these notes as self contained as possible and so all the information needed to read through them is either from an Algebra or Trig class or contained in other sections of the notes. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasnt covered in class.

2. Because I want these notes to provide some more examples for you to read through, I dont always work the same problems in class as those given in the notes. Likewise, even if I do work some of the problems in here I may work fewer problems in class than are presented here.

3. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible when writing these up, but the reality is that I cant anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that Ive not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.

4. This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.

AreaandVolumeFormulasIn this section we will derive the formulas used to get the area between two curves and the volume of a solid of revolution. Area Between Two Curves We will start with the formula for determining the area between ( )y f x= and on the interval [a,b]. We will also assume that

( )y g x=( ) ( )f x g x on [a,b].

We will now proceed much as we did when we looked that the Area Problem in the Integrals Chapter. We will first divide up the interval into n equal subintervals each with length,

b axn =

Next, pick a point in each subinterval, *ix , and we can then use rectangles on each interval as follows.

The height of each of these rectangles is given by,

( ) ( )* *i if x g x and the area of each rectangle is then,

( ) ( )( )* *i if x g x x So, the area between the two curves is then approximated by,

( ) ( )( )* *1

n

i ii

A f x g x=

x The exact area is,

( ) ( )( )* *1

limn

i in iA f x g x == x

Now, recalling the definition of the definite integral this is nothing more than,

( ) ( )ba

A f x g x dx= The formula above will work provided the two functions are in the form and

. However, not all functions are in that form. Sometimes we will be forced to work

with functions in the form between

( )y f x=( )y g x=

( )x f y= and ( )x g y= on the interval [c,d] (an interval of y values). When this happens the derivation is identical. First we will start by assuming that ( ) ( )f y g y on [c,d]. We can then divide up the interval into equal subintervals and build rectangles on each of these intervals. Here is a sketch of this situation.

Following the work from above, we will arrive at the following for the area,

( ) ( )dc

A f y g y= dy So, regardless of the form that the functions are in we use basically the same formula.

Volumes for Solid of Revolution Before deriving the formula for this we should probably first define just what a solid of revolution is. To get a solid of revolution we start out with a function, ( )y f x= , on an interval [a,b].

We then rotate this curve about a given axis to get the surface of the solid of revolution. For purposes of this derivation lets rotate the curve about the x-axis. Doing this gives the following three dimensional region.

We want to determine the volume of the interior of this object. To do this we will proceed much as we did for the area between two curves case. We will first divide up the interval into n subintervals of width,

b axn =

We will then choose a point from each subinterval, *ix . Now, in the area between two curves case we approximated the area using rectangles on each subinterval. For volumes we will use disks on each subinterval to approximate the area. The area

of the face of each disk is given by ( )*iA x and the volume of each disk is ( )*i iV A x x= Here is a sketch of this,

The volume of the region can then be approximated by,

( )*1

n

ii

V A x=

x The exact volume is then,

( )

( )*

1lim

n

in ib

a

V A x

A x dx

=x=

=

So, in this case the volume will be the integral of the cross-sectional area at any x, ( )A x . Note as well that, in this case, the cross-sectional area is a circle and we could go farther and get a formula for that as well. However, the formula above is more general and will work for any way of getting a cross section so we will leave it like it is. In the sections where we actually use this formula we will also see that there are ways of generating the cross section that will actually give a cross-sectional area that is a function of y instead of x. In these cases the formula will be,

( ) ,dc

V A y dy c y= d

y

In this case we looked at rotating a curve about the x-axis, however, we could have just as easily rotated the curve about the y-axis. In fact we could rotate the curve about any vertical or horizontal axis and in all of these, case we can use one or both of the following formulas.

( ) ( )b da c

V A x dx V A y d= =

PrefaceArea and Volume Formulas