How Do You Determine Which Function is Upper or Lower When Integrating?

[fk378]

This is a general question... when given 2 functions (or more) how can you plug in numbers to determine which function is bigger? Let's say x^2 and x^3...if I plug in 2, x^3 would seem to be the bigger function, but when I graph it, x^2 is the bigger function, no?

 

[Dick]

Which is 'bigger' would seem to depend on what x is, wouldn't it?

 

[Gib Z]

If by "bigger" you mean has its function values grow faster, then x cubed grows faster. If by "bigger" you mean how wide the graph looks when you draw it, x squared is bigger.

 

[HallsofIvy]

As both Dick and Gib Z have pointed out, first, you have to define "bigger" for functions!

There are three definitions commonly used: max |f(x)| where the maximum is taken for x in a specific set, \int |f(x)dx where the integral is over a specific measurable set, and \sqrt{\int f(x)^2 dx} where, again, the integral is over a specific measurable set.

Technically, those are referred to as the "uniform", L₁, and L₂ norms, respectively.

In any case, when you are talking about how "big" a function is, you are talking about the y value. You appear to be looking at the "width" of graph- the distance horizontally, from the y-axis. That is exactly the opposite of what we would normally think of as one function being bigger than the other. x² looks "bigger" horizontally than x³ precisely because for a given y, x must be larger in x². That, of course, is because, for a given x, y is larger in x³.

 

[fk378]

Thank you all for your input and excuse me for my faulty use of language.

By which function is "bigger" I actually am referring to which function that is determined to be the top order versus the function that is determined to be the bottom order, when dealing with integrals.

So if given x^2 and x^3, and without graphing it first, how can you tell which function should be designated as top order and which one would be bottom order?

 

[Dick]

As I said, it depends on the value of x. Sometimes x^2 is bigger and sometimes x^3 is bigger. You have to find the points where they cross. Solve x^2=x^3 and set your limits accordingly.

 

[HallsofIvy]

Thank you all for your input and excuse me for my faulty use of language.

By which function is "bigger" I actually am referring to which function that is determined to be the top order versus the function that is determined to be the bottom order, when dealing with integrals.

So if given x^2 and x^3, and without graphing it first, how can you tell which function should be designated as top order and which one would be bottom order?

I think that's worse! What do YOU mean by "top order" and "bottom order"?

I suspect that you are talking about finding the area between two curves and want to know how to determine the upper and lower limits of integration. Don't look for any general rule- the best thing you can do is graph the curves so you can SEE.

If you are integrating with respect to x, then "size" is determined by the y-value. Between x= -1 and x= 1, x³< x². For x> 1 or x> -1, it is the other way around.

If you are integrating with respect to y, "size" is determined by the x-value. In fact, the best thing to do in that situation is to solve each equation for x and treat x as a function of y.