Finding the Area Between Parabolas: Double Integral Help Needed

[Juggler123]

I need to find the area between the parabolas x=y^2 and x=2y-y^2, I know I need to use a double integral but am having difficulty finding the limits. Can anyone help please?

 

[tiny-tim]

Hi Juggler123!

(try using the X² tag just above the Reply box )

I need to find the area between the parabolas x=y^2 and x=2y-y^2, I know I need to use a double integral …

No, for an area you'll only need a single integral …

divide the area into strips of width dy, find the area of each strip, and integrate over y.

 

[HallsofIvy]

Well, you certainly could use a double integral to find area!

The double integral
$\int_{x=a}^b \int_{y= f(x)}^{g(x)} 1 dy dx$
has a very simple first integral of f(x)- g(x) so that the second integral is
$\int_{x=a}^b f(x)- g(x) dx$
and that is the integral for area between the curves that tiny-tim is referring to.

So, for your problem, first, determine for what values of x the graphs intersect. That will be your limits of integration a and b for the "outer" integral (a being the smaller, b the larger, of course). The decide which of the graphs is higher inside that interval. That will be f(x) and g(x), the limits of integration for the "inner" integral.