## Triple integral

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We consider the triple integral $\iiint_E f(x,y,z) \,dV$ where $$E$$ is the solid bounded by $$z = 1-x^2$$, $$z = 1-y$$, $$y = 1$$ and $$x = 0$$. There are six ways to write this triple integral as an interated integral in Cartesian coordinates: $\int_0^1 \int_{1-z}^1 \int_0^{\sqrt{1-z}} f(x,y,z) \,dx \,dy \,dz$ $\int_0^1 \int_{1-y}^1 \int_0^{\sqrt{1-z}} f(x,y,z) \,dx \,dz \,dy$ $\int_0^1 \int_0^{\sqrt{1-z}} \int_{1-z}^1 f(x,y,z) \,dy \,dx \,dz$ $\int_0^1 \int_0^{1-x^2} \int_{1-z}^1 f(x,y,z) \,dy \,dz \,dx$ $\int_0^1 \int_0^{\sqrt{y}} \int_{1-y}^{1-x^2} f(x,y,z) \,dz \,dx \,dy$ $\int_0^1 \int_{x^2}^1 \int_{1-y}^{1-x^2} f(x,y,z) \,dz \,dy \,dx$

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Joseph Lo