06 Feb '09 04:30

The question:

Find the volume of a solid, who's base is bounded by the graph of

y=x^3 , y=0, x=1

and whose cross section is that of a semiellipse(a hieght, twice that of it base) taken perpendicular to the y axis.

solving the problem isn't a problem, Im intentionally making it more difficult that it has to be in hopes of gaining a little insight.

here is how i want to approach it!

find the quadratic equation that has the ordered pair solutions (0,0),(b,0), (1/2*b, 2b)

the equation in what would be the z direction would be

z = -x^2 + 8x

now back to the x-direction, the base of this elipse in terms of "y" must be

base = (1-y^(1/3))

now here is where im not sure i can do what im doing

the hieght in terms of y = 2-2y^(1/3)

so setting

-x^2 + 8x = 2-2y^(1/3)

I arrive at

x = sqrt( 2y^(1/3) + 14 ) + 4

before I go and try to evaluate this integral which i believe would be

INT(0_1) (sqrt( 2y^(1/3) + 14 ) + 4 )*(1-y^(1/3)) dy

I dont even relistically think i can?

does this seem logical, or is it horse crap?

Find the volume of a solid, who's base is bounded by the graph of

y=x^3 , y=0, x=1

and whose cross section is that of a semiellipse(a hieght, twice that of it base) taken perpendicular to the y axis.

solving the problem isn't a problem, Im intentionally making it more difficult that it has to be in hopes of gaining a little insight.

here is how i want to approach it!

find the quadratic equation that has the ordered pair solutions (0,0),(b,0), (1/2*b, 2b)

the equation in what would be the z direction would be

z = -x^2 + 8x

now back to the x-direction, the base of this elipse in terms of "y" must be

base = (1-y^(1/3))

now here is where im not sure i can do what im doing

the hieght in terms of y = 2-2y^(1/3)

so setting

-x^2 + 8x = 2-2y^(1/3)

I arrive at

x = sqrt( 2y^(1/3) + 14 ) + 4

before I go and try to evaluate this integral which i believe would be

INT(0_1) (sqrt( 2y^(1/3) + 14 ) + 4 )*(1-y^(1/3)) dy

I dont even relistically think i can?

does this seem logical, or is it horse crap?