Vector Calculus | Part IA, 2002

The domain SS in the (x,y)(x, y) plane is bounded by y=x,y=ax(0a1)y=x, y=a x(0 \leqslant a \leqslant 1) and xy2=1(x,y0)x y^{2}=1(x, y \geqslant 0). Find a transformation

u=f(x,y),v=g(x,y)u=f(x, y), \quad v=g(x, y)

such that SS is transformed into a rectangle in the (u,v)(u, v) plane.


Dy2z2xdxdydz\int_{D} \frac{y^{2} z^{2}}{x} d x d y d z

where DD is the region bounded by

y=x,y=zx,xy2=1(x,y0)y=x, \quad y=z x, \quad x y^{2}=1 \quad(x, y \geqslant 0)

and the planes

z=0,z=1z=0, \quad z=1

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