Integrating Areas between curves

[Isaiasmoioso]

I can't figure how to solve this problem. Is says ∫∫(x^2)y dA where R is the region bounded by curves y=(x^2)+x and y=(x^2)-x and y=2. I can't figure how to do the limits with that. Please help!

 

[DonAntonio]

I can't figure how to solve this problem. Is says ∫∫(x^2)y dA where R is the region bounded by curves y=(x^2)+x and y=(x^2)-x and y=2. I can't figure how to do the limits with that. Please help!

$y=x^2+x$ is an ascending straight parabola with zeroes at $-1\,,\,0$ , and $y=x^2-x$ is a similar such parabola with zeroes at $0\,,\,1$.

Well, now draw both parabolas together, find their intersection points and...voila!

DonAntonio

 

[Isaiasmoioso]

I was able to get all that. I even graphed it but I can't figure out how to go about writing the integral.

 

[HallsofIvy]

Where do the two parabolas intersect? Where does y= 2 intersect the two parabolas?

It looks to me that you will want to do this in two separate integrals. First, take x from the value of x where the line y= 2 intersects $y= x^2+ x$ to x= 0. You should be able to see that, for each x, y goes from the parabola up to $y= x^2+ x$. Then do the second integral from x= 0 to the intersection of y= 2 and $y= x^2- x$.

 

[vish22]

Hey man,you should do this in 2 double integrals-
0 2
∫ ∫ f(x,y) dy dx
-1 (x(x-1))

1 2
∫∫ f(x,y) dy dx
0 (x(x+1))and then add the 2 for your total volume.
I hope this is right