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Calculus/Area between 2 curves

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Expert: - 10/12/2008


Question
Hi, I'm trying to find the area between 2 curves enclosed by y=x and y=x^3. When I solve for x I get x=-1,0,1.  So, would I make the x in [-1,1] which is obviously equal to 0 or do I make x in [-1,0] or [0,1]? What is the general rule when you have x equal to more than 2 answers as in this example.

Thank you

Answer

Area between curves
Questioner:   Tyrene
Category:  Calculus
 
Subject:  area between 2 curves
Question:  Hi, I'm trying to find the area between 2 curves enclosed by y=x and y=x^3. When I solve for x I get x=-1,0,1.  So, would I make the x in [-1,1] which is obviously equal to 0 or do I make x in [-1,0] or [0,1]? What is the general rule when you have x equal to more than 2 answers as in this example.

Thank you
.............................
Hi, Tyrene,

When you draw the graph (see attached) you recognize the difficulty. (Sorry, I mean, you recognize the issue.  "Difficulty" is so 20th century.)  

You have two pieces of area.  In one of them, y = x is above and in the other it is below.  So if you just do:

{1
| (x - x^3) dx
}-1

OR

{1
| (x^3 - x) dx
}-1

you are going to get zero; the two pieces will cancel.  Here is a Theorem you will come up against later:

{a
| (any odd function(x)) dx = 0
}-a

and x^3 - x is surely an odd function.

Now that you see the difficulty, you can deal with it. Just do:

{1
| (x^3 - x) dx
}0
OR
{1
| (x - x^3) dx
}0

and when you get the answer, double it.  [And be sure to 'positive' it in case you got a negatve answer.]

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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(See above.)

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