Posted by T.Gracken on November 06, 2002 at 20:13:40:
In Reply to: Re: areas between curves posted by Joel on November 06, 2002 at 18:48:17:
: : does this appear correct?
: : find the area between the following functions...
: : y = x^3-x
: : y = 3x
this makes no sense. there is an infinite amount of area between the curves. Do not para-phrase. Is the actual question as stated or is it closer to "determine the area bounded by the curves...?" Although you may not see or interpret the difference, the English language is harsher than math. so... we have a non-math problem. that is, what language do you want us to interpret this in?
O.K. we'll assume some things... like you want the area bounded by the curves.
as Joel mentioned (paraphrasing); it would be silly to think the answer is 0 just because the function is odd (even, etc.)
so consider the question at hand. If you want an area you wish to find a positive valued number relating to how many squares it would take to completely cover a flat surface. If you want a Riemann sum then the answer might be 0 but that is a different problem...
the area can be found using basic (definite) integrals.
I believe you have the basic idea and the basic approach, but you need to develop your fundamentals a bit further
: : first find points of intersection...3x=x^3-x ---> 0 = x^3-4x ---> 0 = x(x^2-4) ---> 0 = x(x+2)(x-2) so points of intersection are (0,0) (-2,-6) (2,6)
: : Then I graph...the areas between the curves are clear...
: : According to the "Integrals of Symmetric Functions", since both functions are odd, they will net out to zero...which means my net area between the curves is also zero....
: : A = S f(x) - g(x)
: : Is this correct?
: I don't think so. That argument simply defies reason. (Of course, so do some of the other arguments we have been taught in calc, but that just goes too far for me.)
: That symmetry rule tells you that the definite integral of an odd function individually from -a to a is zero. But to calculate areas BETWEEN TWO functions, my eyes tell me that those two lines are marking out an area that is measurable and it isn't zero. That's like saying that a circle drawn around the origin has zero area, but if you slide it up r units along the y axis, area magically appears. When we have any other two curves that mark out areas that overlap the axes, we add the areas, essentially taking the "absolute values" of the areas. It makes no sense to do it differently just because the curves are symmetrical. What would you do if one curve was odd and the other was even, or not symmetrical at all?
: Anyway, enough of my diatribe. I say you should go ahead and compute 2 areas:
: the integral from -2 to 0 of x^3 - 4x
: and the integral from 0 to 2 of 2x - x^3
: and then add them together for the total area
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