Calculus Lab #5:  Integration & Area

Goals:	1.  Learn how to evaluate indefinite and definite integrals in Mathematica
	2.  Learn how to check solutions to integrals
	3.  Practice calculating areas under curves

0.  Why should we ever calculate integrals by hand?

The process of finding antiderivatives of elementary functions is often called integration.  In a traditional calculus textbook, entire chapters are devoted to learning various techniques for finding antiderivatives.  You will learn these techniques in MATH 192 when you study chapter 7.
	Today, software packages such as Mathematica (or websites or TI-89s) are capable of finding antiderivatives in nearly all cases where it is possible to do so.  For this reason, the ability to compute difficult integrals by hand is considerably less important than it used to be, and many textbooks and courses now devote less time to this topic than the traditional approaches.
	It may be tempting to leave antidifferentiation entirely to the computer (it's definitely tempting to me).  However, there are four good reasons for learning how to find antiderivatives by hand:

1.  The capability to do basic calculations mentally is fundamental to thinking about mathematics. If you are unable to do basic calculus operations in your head, you will have difficulty thinking about calculus and using it in other contexts.

2.  Although computers seldom make errors, people make mistakes all the time when working with computers. If you are unable to catch those mistakes, your work will be worthless. Knowing how to work problems by hand is the single most important way to catch mistakes on the computer.

3.  To be an effective problem solver, you must have a variety of tools to work with, and learn to use them appropriately. The best solution to the problem of computing antiderivatives is to learn to find them both by hand and using the computer. Then you can use whichever method is most effective or convenient in a given situation.

4.  Antidifferentiation is not as straightforward as differentiation; there are often choices to be made, and software will often not make the choices that you or I would make. Answers obtained by software are rarely incorrect, but they are often in unfamiliar or undesirable forms. For example:

∫(1/(Sqrt[1 + (x^2)])) \[DifferentialD]x

Unless you are familiar with the inverse hyperbolic sine function arcsinh(x), you would probably prefer this answer: log ((Sqrt[1 + (x^2)])  +  x).

FullSimplify[(∂_x) Log[(Sqrt[1 + (x^2)]) + x]]

Of course there are many times when using a computer is the better choice. If you need to find the antiderivative of (x^4) (e^(-2 x)) sin (3  x), by all means take advantage of the computer to do so! If you are working a long problem or project on the computer, you may want to do all antiderivatives (even easy ones) on the computer to make the results available for further calculations.

1.  Indefinite Integrals in Mathematica.

The easiest way to enter antiderivatives in Mathematica is to use the BasicInput palette.  Make sure you tell Mathematica which variable you're integrating.

∫(x^2) \[DifferentialD]x

∫((x^2) + 
            4  x) \[DifferentialD]x

Mathematica almost always leaves integrals in complex forms.  I recommend you use the command FullSimplify whenever you find an integral, so you're sure you have the most reduced form of the answer.

FullSimplify[∫((x^2) + 
              4  x) \[DifferentialD]x  ]

How can we check to see if the answer is correct?

FullSimplify[(∂_x) ((1/3) (x^2) (6 + 
                x))]

Some indefinite integrals are deceptively complex...

∫(Sqrt[Tan])[x] \[DifferentialD]x
FullSimplify[∫(Sqrt[Tan])[x] \[DifferentialD]x]

2.  Definite Integrals in Mathematica.

Definite integrals are just as easy to evaluate.  Evaluate the following definite integral by hand and verify the answer provided by Mathematica.

(∫_0^((π/2)))( 
          Sin[x] Cos[x]) \[DifferentialD]x

(∫_2^12) ((x^2) + 
            4  x) \[DifferentialD]x  
N[(∫_2^12) ((x^2) + 
              4  x) \[DifferentialD]x ]

3.  Practice Questions (Turn in for credit)

Use Mathematica to evaluate the following integrals:
Page 307:  3, 5, 7, 11, 13
Page 308:  27, 35, 39, 59, 65, 67

5.  Questions to turn in

1.  Turn in your answers to the practice questions in section #3

6.  Reflection

At the end of each Mathematica session, I want you to write a sentence or two describing what you learned (or just list important concepts from this lab).  You can later use these reflections as a reference for other Mathematica Labs.

Print out your answers to the lab questions and turn them in for credit.