We are now within reach of fulfilling our goals stated at the outset of my Introduction to Integral Calculus. These goals are to derive the following:
(a) Formulae for volume of the cylinder, cone, sphere, and various parabaloids using two methods; the slab (disk) method and the shell method.
(b) Formulae for surface area of the cone and sphere using similar methods to those mentioned above.
(c) Formulae for area of the circle and ellipse.
(d) Formula for arc length of a circle given its central angle. For this, we will reference the Mean Value Theorem and our knowledge of differentiaion.
Before proceeding any further, however, a "new" method of integration must be introduced, that being Integration by Trigonometric Substitution; this was eluded to in the previous set of notes. The example illustrated on pages 20 & 21 in my notes on Integration by Partial Fractions revealed that the antiderivative of the expression "[3/(x^2+1)]" was "[3arctan(x)]". This method and the reasoning behind it will be explained in my next set of notes; we will also be made aware of why the "Natural Logarithm Method" WILL NOT work for this expression. My notes on Integration by Trigonometric Substitution can be found in the link directly below.
Integration by Trigonometric Substitution - Samuelson
Visit the links below for additional information on this method of integration. The first one takes you to a lecture given by a guest instructor at MIT.
Trigonometric Substitution & Polar Coordinates
Integration by Trigonometric Substitution (1)
Integration by Trigonometric Substitution (2)
Integrating Algebraic Functions
My Belief Statement
My Belief Statement: _It is better to be defeated on principle than to win on lies._ Arthur Calwell
Many students have a serious phobia when it comes to Math class, let alone writing an exam in that subject. I believe this phobia can be dispelled by "digging" into the "WHY " and the "HOW". In fact, many students WANT this. I attempt, wherever possible, to connect a geometric representation to its algebraic counterpart; an analysis of how the two versions interact is important. Contrary to what SOME individuals may think, I'm not at all afraid of technology. I AM, however, cautious to not "jump into the deep end"; we need to grow with this and "dovetail" the old with the new so that they COMPLIMENT each other.
I have several links in the sidebar at the right, as well as some of my own discriptions on various topics. These will grow in number and evolve over time, as will my hand-written notes; these can be viewed by following appropriate links which you will find throughout this blog. These notes will also be posted on my facebook site if that happens to be your preferred mode of viewing; I've included a link to that in the sidebar as well.