Expert: Paul Klarreich - 10/1/2009

Question
Hello,

I am working in Calculus 2 on the applications of integration.  It deals with finding the volume/area through the utilization of the integration method.

My question is:  Find the area of the region enclosed by the curves x= y^2 - 2 and x = |y|.  

I know you have to sketch the graph.  I set the two equations equal to each other to find the endpoints, but I am unclear whether I need to "flip" these equations from reading "x" instead of "y".  If you are unclear on this just email me back.

Thanks,
Andrew

Answer
Questioner: Andrew Prusack
Country: United States
Category: Calculus
Private: No
Subject: Applications of integration...
Question: Hello,

I am working in Calculus 2 on the applications of integration.  It deals with finding the

volume/area through the utilization of the integration method.

My question is:  Find the area of the region enclosed by the curves x= y^2 - 2 and x =

|y|.  

I know you have to sketch the graph.  I set the two equations equal to each other to find

the endpoints, but I am unclear whether I need to "flip" these equations from reading "x"

instead of "y".  If you are unclear on this just email me back.

Thanks,
Andrew
.........................................
Hi, Andrew,

Yes, you could easily 'flip' the equations.  But it is not necessary.   All you have to do is:

1) Integrate in the y-direction.  (Perfectly legal.)

2) Use only the top part -- y goes  0 to 2 -- and double it. (a common, if sneaky, trick.)

So it goes:

{y=2
|    [ y - (y^2 - 2) ] dy      << use |y| = y for positive y.
}y=0

and then double the answer.

I get 20/3 at the end.