If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page.

QuickTime free download.

	THE INTEGRATOR  is a free function integrator.  You need to read their instructions on entering input to use it effectively.  It is from Wolfram Research and powered by Mathematica.
	You can also compute integrals using Derive on your computer in class to check your answers or you can use the antiderivative feature on the Vanderbilt Toolkit.
	Here is a table of integrals.
	Hotmath  You can look at free solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, eighth edition.  They now have chapters 1 - 12 available.  Only a few solutions are still free.
	UC Davis has a site with a lot of problems and worked out solutions dealing with many topics in Calculus I and Calculus II including some of the techniques of integration we are studying in this chapter.
	Maple commands to factor and to perform a partial fraction decomposition

 8.1   You will need to be able to apply the basic integration rules reviewed in the section and the procedures demonstrated on page 521 of your text.

8.2   You will need to be able to demonstrate the application of the Integration by Parts technique.  Here is the derivation of the Integration by Parts formula.  These are additional links to tutorials and examples demonstrating integration by parts:  SOS Integration by Parts, HMC on Integration by Parts.  Here are some drill  problems with step by step solutions:  Drill set 1, drill set 2, drill set 3.

8.3   You will need to be able to evaluate the types of trigonometric integrals in this section.  Here are links to some tutorials on trigonometric integrals:  Powers and products of sines and cosines, more products of sines and cosines, other trig powers, rational expressions of trig functions.

8.4   You will need to be able to demonstrate the use of the trigonometric substitutions presented in this section and summarized on page 543.  Here is what Karl's Calculus Tutor has to say about trigonometric substitution and here is HMC on trig substitutions.  You will need to complete the square in computing some of the integrals in this section.  Here are some drill problems involving trigonometric substitution.

8.5   You will need to be able to demonstrate the use of partial fraction decomposition to evaluate the integral of a rational function.  These are additional links to tutorials and examples demonstrating integration by partial fraction decomposition:  SOS on Rational Functions, Karl on partial fraction decomposition, HMC on partial fractions.  Here are some examples and drill problems:  Examples, drill problems 1, drill problems 2.

8.6   You will need to be able to use a table of integrals or a computer algebra system in evaluating indefinite integrals.

8.7   You will need to be able to recognize indeterminate forms and use L'Hopital's Rule to evaluate a limit involving an indeterminate form when appropriate.  In the figure on the right are the graphs of y = sin(7x)/(3x) (red) and y = 7cos(7x)/3 (green).  We can apply L'Hopital's Rule to find the limit as x approaches zero of sin(7x)/(3x) and compute the limit by finding the limit as x approaches zero of 7cos(7x)/3.  Why would this not work (and be unnecessary) in finding the limit as x approaches 1 of sin(7x)/(3x)?  Would applying L'Hopital's Rule incorrectly ever yield the correct result in this example?  Here are some examples from The University of Tennessee's Visual Calculus:  Drill on L'Hopital's Rule with finite limits, drill on L'Hopital's Rule with infinite limits.

8.8 You will need to be able to compute the limits necessary in evaluating improper integrals that converge and will need to be able to demonstrate that some improper integrals diverge.  This will involve both improper integrals with infinite limits of integration and improper integrals with infinite discontinuities.  You should be able to prove and apply Theorem 8.5, page 584, involving a special type of improper integral.  Here are some drill problems on improper integrals:  Drill problems 1, drill problems 2.  

Take Home Problems

1.  Integrate the function pictured in the figure on the left below with limits of integration from 0 to 4.  This is an improper integral so set it up in terms of limits.

2.  Integrate the function pictured in the figure in the center below with limits of integration from 0 to 3.  This is an improper integral so set it up in terms of a limit.

3.  Find the volume of Gabriel's horn (see figure below on the right, click on it to see an animation, and Example 11, page 584 in your text) and show that the surface area is infinite.  Maple Picture

Gabriel's Horn DPGraph Picture1      DPGraph Picture2

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats