5

Given a question like this:

Find the area between ${y = x^2 + 2x - 3}$ and ${y = 2x^2 -5x -3}$.

I know how to find the area ${\int y_1 - y_2}$ but how can I tell which one is the top curve? Are there any shortcuts to determining the top curve?

dagda1
  • 825
  • How do you know any one is on top of the other one? The curve might intersect, thereby changing roles in this respect. Apart from that: if $y_2-y_1 > 0$ then $y_2$ is on top... – Thomas Feb 06 '16 at 06:47
  • If they don't cross you can simply integrate and take the absolute value. – Henricus V. Feb 06 '16 at 06:48

2 Answers2

5

First find the points of intersection $$x^2+2x-3=2x^2-5x-3$$ $$x(x+2)=x(2x-5)$$ $x=0$ or $x+2=2x-5$

$x=0$ or $x=7$

As the curves are continuous, the simplest way is to take any point between $0$ and $7$ and find for which the value of $y$ is greater.

If the curves intersect more than twice, do this for every two consecutive points of intersection.

And finally, it does not matter much, when you are finding between two points with constant value (and if the functions don't intersect more than twice). If you get the order wrong, all that happens is that you get the negative of the actual answer. Taking the modulus will give you the area.

GoodDeeds
  • 11,185
  • 3
  • 22
  • 42
3

Assuming you work only with polynomials and they don't intersect here is a little shortcut:

Find the highest exponent, that is different for both polynomials. The one, with the higher coefficient is on top.

wotanii
  • 131