Find the average value of the function in the area.

Question

I have such function $f(x,y,z)=e^{\sqrt{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}}}$ and the area $ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\leqslant 1$. Actually the main problem is the graph. I used Geogebra to plot this kind of area, but didn't succeed. Can anyone give a hint and tell me how to plot it or what it will look like? And then how to proceed in calculating the integral To calculate the integral I think we should use $$x = a\cdot r\cdot cos(\varphi)cos(\psi)$$ $$y = b\cdot r\cdot sin(\varphi)cos(\psi)$$ $$z =c \cdot r\cdot sin(\psi)$$

Answer 1

The map $$g:\quad(u,v,w)\mapsto(x,y,z):=(au,bv,cw)$$ maps the unit ball $B$ of $(u,v,w)$-space bijectively onto the ellipsoid $E$ which is your integration domain in $(x,y,z)$-space. Since the Jacobian of $g$ computes to $J_g(u,v,w)\equiv abc$ we obtain $$Q:=\int_E e^{\sqrt{(x/a)^2+(y/b)^2+(z/c)^2}}\>{\rm d}(x,y,z)=abc \int_B e^{\sqrt{u^2+v^2+w^2}}\>{\rm d}(u,v,w)\ .$$ Since the integrand on the RHS is rotationally symmeric we partition $B$ into infinitesimal spherical shells of volume $4\pi r^2\,dr$ and then obtain $$Q=4\pi abc \int_0^1 e^r\>r^2\>dr=4\pi abc(e-2)\ .$$

Answer 2

The volume is an ellipsoid like the one in the picture below.