## Integration by Trigonometric Substitution

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)

Thanks for sharing your insights on Integration by Trigonometric substitution. I am now more humble, yet better informed.

Tom Humphreys

Thanks for the nice words Tom; not sure why you’re more humble though…….