Calculus 2 - Mathematics 120 - Spring 2008

Lab problem: finding the length of an arc on a parabola

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1. Finding the length of an arc on a parabola.
(a) Write a definite integral for the length of the arc on the parabola y = x^2 from x= 0 to x = 1.
(b) Use a spreadsheet to approximate this definite integral using n = 10, 100, 1000 midpoint rectangles.
(c) Use Mathematica to find the answer accurate to five decimal places. Here's a brief intro to Mathematica (also pasted below).
(d) Compare your results from parts (b) and (c). How many midpoint rectangles are enough to give an accuracy of five decimal points?


Integrals via Mathematica

  1. Click:  Start > All Programs > Mathematics > Mathematica.

  2. Once Mathematica is open, click:  File > Palettes > Basic Input.

  3. In the Palette that opens (or may already have been open), you will see various buttons, such as for square root, and definite or indefinite integrals:

                        
     

  4. You can click on and type inside each of the "input boxes"  (or you can use Tab and Shift-Tab to jump between the boxes).

  5. When done typing in the desired expressions, press Shift-Enter, or the Enter key that's on your keyboard's number-pad.

  6. When you try to find a definite integral, Mathematica usually gives you a long expression instead of just one number as the answer (try it for the problem you're working on!). Here's what's happening:

    Type in Sqrt[2] (or use the Basic Input palette) to find the square root of 2; press Shift-Enter. Do this and see what happens!
    What's the result? Mathematica doesn't give you 1.4142...; instead it gives you back what you typed in! Why? Because it prefers to give you an exact answer. But you can tell it to give you a numerical approximation by typing in N[Sqrt[2]] ; try it!  N[ ] means "give me a Numerical answer".
    Note: capitalization and bracket-type matter: It's N[ ], not n[ ], nor N( ).
     

  7. Now go back to the problem you were working on and put N[ ] around it and press Shift-Enter.

 

 

Updated: 31 August, 2009 17:44:19