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

I know it is possible to do it like so:

[itex]\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][/itex]

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

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???

thanks for your help, b.

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# Can you verify this please?

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