calculate $\textrm{vol}(E)$ for $E=\{(w,x,y,z)|0<w<1-(x^2+y^2+z^2)^3\}$

Question

Calculate $\textrm{vol}(E)$, for $$ E=\{(w,x,y,z)|0<w<1-(x^2+y^2+z^2)^3\}. $$

My try:
Exchange for : $0<w+(x^2+y^2+z^2)^3<1$

Integrate first by range of $[0,1] dw$, and get $$0<1/2+(x^2+y^2+z^2)^3<1$$ Now, replace for polar, and get the integral: $$ \int ^{ \pi}_{0}\int ^{2 \pi}_{0} \int_{0} ^{\sqrt[3]{0.5}}0.5R^2+R^8)\sin(\phi)\; dr d\theta d\phi $$ which leads to $$\int ^{ \pi}_{0}\int ^{2 \pi}_{0} (0.25/3+1/72)\sin(\phi)\; dr d\theta d\phi $$ which is cleary wrong as it leads to $$ 4\pi^2\cdot\frac{7}{72}=\pi^2\cdot\frac{7}{18}$$ which is clearly wrong. Does anyone spot my mistake?

Answer 1

I tried $$\int_{-1}^1\left(\int_{-(1-x^2)}^{1-x^2}\left(\int_{-(1-x^2-y^2)}^{1-x^2-y^2}\left(\int_0^{1-(x^2+y^2+z^2)^3}{\rm d}w\right){\rm d}z\right){\rm d}y\right){\rm d}x$$ and, with my favorite CAS, found the stupid answer $$I=\frac{1541257128064}{619377984225}$$ I know...

Answer 2

Whenever I try to find volumes of higher dimensional objects I try to refer to a simpler case. How would you find the volume of the solid defined by $0<z<1-(x^2+y^2)^3$? It equals $\int_{x^2+y^2<1}(1-(x^2+y^2)^3)dA$. Similarly Vol$(E)=\int_{x^2+y^2+z^2<1}(1-(x^2+y^2+z^2)^3)dV$ where $E$ is the solid in your problem. We can evaluate this integral easily by converting to spherical coordinates. This is the same answer Nicolas FRANCOIS would have gotten if he remember to put in square roots in the bounds for the middle two integrals ;) Click here to see this.