The curves intersect at (0,0), (1,1)

Translate the x-axis to coincide x=-1 as the new X-axis:

x->X

y->Y-1

the curves change correspondingly:

y=x^2->Y=x^2+1

x=y^2->X=(Y-1)^2

=>the new intersecting points are (0,1), (1,2).

=>the volume of revolution is

........1

V=piS[(sqr(x)+1)^2-(x^2+1)^2]dx=

........0

...1

piS[x+2sqr(x)-x^4-2x^2]dx=

...0

......1

pi[(x^2)/2+4(x^1.5)/3-(x^5)/5-2(x^3)/3]

......0

=29pi/30

=3.04 approximately.

y= x^2

x=sqrt(y)

x= y^2

sqrt(y)=y^2

y=y^4

y^4-y=0

y^3(y-1)=0

y=0; y=1 (limits of integration)

Volume = pi Int ((sqrt(y)+1)^2 - (y^2+1)^2) dy

Volume = pi Int (-y^4-2y^2+y+2sqrt(y)) dy

= 29pi/30

http://www.wolframalpha.com/input/?i=graph+x%3Dy%5...

The "shells" method will be convenient. The intersections of the two curves are at (0,0) and (1,1).

Height of a shell = sqrt(x) - x^2.

Radius of a shell = x - (-1) or x+1.

Thickness of a shell = dx.

Volume of a single shell = 2*pi*R*H* dx

= 2*pi*[x^(3/2) - x^3 + sqrt(x) - x^2] dx.

Indefinite integral is

2*pi*[(2/5)x^(5/2) - (1/4)x^4 + (2/3)x^(3/2) - (1/3)x^3].

Plug in 1 and 0, you get

2*pi[24/60 - 15/60 + 40/60 - 20/60] = 29pi/30.