# Can you verify this please?

## Main Question or Discussion Point

Find the volume of the region whose base in the first quadrant of the x-y plane is bounded by $y = x^4$ and $y = \sqrt[4]{x}$, and which is bounded from above by $z = xy^3$

I know it is possible to do it like so:

$\left[\int_{x=0}^{1}\int_{y=0}^{\sqrt[4]{x}}xy^3 dxdy\right] - \left[\int_{x=0}^{1}\int_{y=0}^{x^4}xy^3 dxdy\right]$

but can I do it such: $\int_{x=y^4}^{\sqrt[4]{y}}\int_{y=x^4}^{\sqrt[4]{x}}xy^3 dxdy$

I arrive at the problem of subbing in the limits. I'm not sure if they're correct but by the looks of it to me, the limits of both integrals are dependent on the other variable so I don't know which one to do first???

Mute
Homework Helper
but can I do it such: $\int_{x=y^4}^{\sqrt[4]{y}}\int_{y=x^4}^{\sqrt[4]{x}}xy^3 dxdy$

I arrive at the problem of subbing in the limits. I'm not sure if they're correct but by the looks of it to me, the limits of both integrals are dependent on the other variable so I don't know which one to do first???
That's your clue that you can't do the integral that way. You can't integrate over y and still have a y in the limits of the x integral, since y should have been integrated out.

HallsofIvy
$$\int_{x=0}^1\int_{y= x^4}^{^4\sqrt{x}} xy^2 dydx$$