Triple Integral

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We consider the triple integral \[\iiint_E f(x,y,z) \,dV\] where \(E\) is the solid in the first octant bounded by \(z = 1-x^2\), \(z = 1-y\) and \(y = 1\). There are six ways to write this triple integral as an interated integral in Cartesian coordinates: \[\int_0^1 \int_{1-z}^1 \int_0^{\sqrt{1-z}} f(x,y,z) \,dx \,dy \,dz\] \[\int_0^1 \int_{1-y}^1 \int_0^{\sqrt{1-z}} f(x,y,z) \,dx \,dz \,dy\] \[\int_0^1 \int_0^{\sqrt{1-z}} \int_{1-z}^1 f(x,y,z) \,dy \,dx \,dz\] \[\int_0^1 \int_0^{1-x^2} \int_{1-z}^1 f(x,y,z) \,dy \,dz \,dx\] \[\int_0^1 \int_0^{\sqrt{y}} \int_{1-y}^{1-x^2} f(x,y,z) \,dz \,dx \,dy\] \[\int_0^1 \int_{x^2}^1 \int_{1-y}^{1-x^2} f(x,y,z) \,dz \,dy \,dx\]

Joseph Lo, 2 April 2013, Created with GeoGebra