Is there a trick to tell which is the higher curve?

[Alexandra Fabiello]

The title space is too short for what I want to ask, so here it is.

When determining the area bounded between two functions/curves, you have to know which is the upper curve and which is the lower curve to solve the problem.

Currently I only know how to do it by graphing, but I was wondering if there's another way to do it that's shorter.

 

[member 587159]

The title space is too short for what I want to ask, so here it is.

When determining the area bounded between two functions/curves, you have to know which is the upper curve and which is the lower curve to solve the problem.

Currently I only know how to do it by graphing, but I was wondering if there's another way to do it that's shorter.

When you consider two functions f and g, you can consider the function h = f - g.

h>0 => f>g
h<0 => f<g
h= 0 => f=g

 

[Alexandra Fabiello]

When you consider two functions f and g, you can consider the function h = f - g.

h>0 => f>g
h<0 => f<g
h= 0 => f=g

Thanks!

 

[BreCheese]

To tell which function is "the upper curve" or "lower curve" you can choose any value of x (provided that your choice of x is in the interval of interest) and evaluate each function at that x value to determine which function is the upper (or lower) curve. The upper curve will be the function that yields the greatest number. The function that yields the lowest number (when evaluated at the same value of x) will be your lower curve.
Example. If you are given two functions, f(x)= 2x and g(x)=x^2, and you were asked to calculate the area between the two curves in the interval (0,2), then you could evaluate each function (at any value of x in between x=0 and x=2) and the function that yields the greatest number is the "upper function." 1 is in between (0,2) so evaluate each function at x=1. f(1)=2, and g(1)=1. Therefore, f(x) is your upper function in the interval (0,2). Be careful though because if the interval was beyond x=2 then the upper function changes due to the two functions intersecting. f(x)=2x and g(x)=x^2 intersect at x=2. Try evaluating each function at a value of x in the interval (2,6). For instance, if these functions are evaluated at x=4, then f(4)=8, and g(4)=16. Notice that g(x) yielded the greatest number now, so g(x) is now the upper function in the interval (2,6).
As a last note, its always good to check if the given functions intersect in a given interval before naming each function as "upper" or "lower." If they intersect in the interval, then you will have to evaluate the integral by splitting up the interval, and thus evaluate two integrals (or possibly more if your two functions intersect more than once in the given interval. (If you have questions on how to do this, let me know). (you can check to see if two functions intersect in a given interval by setting them equal to each other and solving for x. If the x you find is in the interval, then the two functions intersect within that interval).

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[Cosmophile]

The ways listed above are the simplest, in my opinion. Though, there is also a handy theorem to know:

Let $f, g$ be two continuous, differentiable functions on some interval $[a,b].$ If $f(a) \leq g(a)$ and $f'(x) \leq g'(x)$ on $(a,b)$, then $f \leq g$ on that interval.

In other words, if $g$ is greater than or equal to $f$ at $a$, and if $g$ grows faster than $f$, then $g$ is bigger than $f$ for all $x > a$ in the interval.

There have been cases where, just by knowing this, I've been able to jump straight into the computation of the integral without having to evaluate each function at some test point. Especially if the interval goes from $[0, b]$. Usually, though, the ways listed above are the best.

 

[zinq]

Given two curves y = f(x) and y = g(x) on some interval a ≤ x ≤ b, then — unless you know in advance that one curve is greater than the other — they may intersect in some number n of points x_j, say for 1 ≤ j ≤ n, between a and b. To find those points, you would need to solve the equation

f(x) = g(x)​

for those values of x satisfying the equation and lying between a and b. Then, assuming you are finding the area per se (and not the "signed area"), you would want to integrate the absolute value

|f(x) - g(x)|​

between each successive pair of x-values x = a, x = x₁, x = x₂, ..., x = x_n, x = b.

To do this, you could just test one value in the interior each of the intervals of integration to see whether f(x)-g(x) is positive or negative. (It will not be 0; do you see why?) That will tell you whether

|f(x) - g(x)| = f(x) - g(x)​

or

|f(x) - g(x)| = g(x) - f(x).​

And finally, add the results together to get the total area between the curves.