Question

Use spherical coordinates to evaluate the triple integral ∭ee−(x2 y2 z2)x2 y2 z2−−−−−−−−−−√dv, where e is the region bounded by the spheres x2 y2 z2=1 and x2 y2 z2=9.

Answer 1

The evaluation of the integral is 2π ($e^{-1} -e^{-9}$)

Given,

$\int\limits^_$$\int\limits^_$$\int\limits^_$$E$$\frac{e(-(x^{2} +y^{2}+z^{2})) }{\sqrt{x^{2} +y^{2} +z^{2} } } \sqrt{dV}$

E is the region bounded by the spheres which is,

$x^{2} +y^{2} +z^{2} =1\\x^{2} +y^{2} +z^{2}=9$

In spherical coordinates we have,

x = r cosθ sin ∅

y = r sinθ sin∅

z = r cos∅

dV = $r^{2}$sin∅ dr dθ d∅

E contains two spheres of radius 1 and 3 ($\sqrt{9}$) respectively, the bounds will be like,

1 ≤ r ≤ 3

0 ≤ θ ≤ 2π

0 ≤ ∅ ≤ π

Then,

$\int\limits^_$$\int\limits^_$$\int\limits^_$E$\frac{e(-(x^{2} +y^{2}+z^{2})) }{\sqrt{x^{2} +y^{2} +z^{2} } } \sqrt{dV}$

$\int\limits^2_0\pi$$\int\limits^\pi _0$$\int\limits^3_1$$\frac{e^{-r^{2} } }{r} r^{2}$ sin∅ dr d∅ dθ

= 2π $\int\limits^\pi _0$$\int\limits^3_1$$re^{-r^{2} }$dr d∅

= -π $\int\limits^\pi _0$$e^{-r^{2} }$$\int\limits^3_1$d∅

= 2π ($e^{-1} -e^{-9}$)

 Learn more about integrals here: brainly.com/question/19234614

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