samuelson mathxp

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 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 will be explained in my next set of notes; we will also be made aware of why the “Natural Logarithm Method” WILL NOT work in finding the antiderivative of 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