How to Identify Functions in Problems Involving Two Curves

[NIZBIT]

I am trouble with problems that don't list each function f(x) or g(x). The book sets them equal to y. Every time I do these I'm getting them wrong. When I check the solution, I'm mistaking f(x) for g(x) or vise versa. So is there a way to tell which y is f(x) or g(x)?

 

[suspenc3]

An example might help be helpful.

 

[radou]

I am trouble with problems that don't list each function f(x) or g(x). The book sets them equal to y. Every time I do these I'm getting them wrong. When I check the solution, I'm mistaking f(x) for g(x) or vise versa. So is there a way to tell which y is f(x) or g(x)?

I can't say I understood your question, but referring to the title of the thread, the area between two 'well-behaved' functions f and g on the interval [a, b] is given with $$\int_{a}^b(f(x)-g(x))dx$$, if f(x) >= g(x), for every x from [a, b].

 

[HallsofIvy]

Of course, the region "ends" where the two curves cross- that is, they have the same y value for that x value. That's why your book "sets them equal to y"- to determine the y values where they cross and thus the limits of integration.
A little hint- if you area comes out negative, then you have the two curves in the wrong order!

 

[NIZBIT]

This is what I'm talking about:

Find the area between these two curves-
y=x^2-4x+3
y=3+4x-x^2

Now how do you know which is f(x) or g(x)?

 

[HallsofIvy]

This is what I'm talking about:

Find the area between these two curves-
y=x^2-4x+3
y=3+4x-x^2

Now how do you know which is f(x) or g(x)?

What? Neither is "f(x)" or "g(x)" until you name them! Call whichever you want f and the other g. It might be that your textbook is using a convention that "f" is always the "upper" curve and "g" is always the "lower" curve so that the area is given by $\int (f(x)- g(x))dx$. If that is the case, then determine which is above the other. Hint: the first is a parabola opening upward, the second a parabola opening downward. You need to decide which is the "upper" curve and which the "lower" curve. You will also need to determine where they intersect.

 

[NIZBIT]

I swear when I worked that problem out it only worked one way. Off the top of my head I had f(x) switched with what the book said. My way was wrong, but then I switched my f(x) for the books way I got the problem right. Had someone double check my math both times and it was good. Thanks for the help!