91影视

We are given,

${鈭�}_{-3}^3{鈭�}_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}{鈭�}_{-\sqrt{9-x^2-y^2}}^{\sqrt{9-x^2-y^2}}f(x,y,z)dzdydx$

By the definition of triple integral ${鈭�}_{a_1}^{a_1}{鈭�}_{b_1}^{b_2}{鈭�}_{c_1}^{c_2}f(x,y,z)dzdydx$represent the volume of the solid region$\mathrm{鈩�}=\left\{(x,y,z)鈭�a_1鈮�x鈮�a_2,b_1鈮�y鈮�b_2,c_1鈮�z鈮�c_2\right\}$

Using this definition, we get

The given iterated integral ${鈭�}_{-3}^3{鈭�}_{\sqrt{9-x^2}}^{\sqrt{9-x^2}}{鈭�}_{-\sqrt{9-x^2-y^2}}^{\sqrt{9-x^2-y^2}}f(x,y,z)dzdydx$represents the volume of the sphere with radius $3$and centered at origin.

Since from the integral limits we observe that localid="1650347847843" $z=\sqrt{9-x^2-y^2}$to $z=\sqrt{9-x^2-y^2}$

Squaring on both sides,

$z^2=9-x^2-y^2$

Simplifying and rearranging, we get

$x^2+y^2+z^2=3^2$

This represents the equation of sphere with radius $3$and centered at origin

Hence the given iterated integral represents the volume of the sphere with radius $3$and centered at origin.