use spherical coordinates to evaluate the triple integral of
(e^-(x^2+y^2+z^2))/(sqrt(x^2+y^2+z^2)dV
where E is the region bounded by the spheres
x^2+y^2+z^2=49 and
x^2+y^2+z^2=81???
Integrate
exp(-r^2)/r 4 pi r^2 dr from r = 7 to r = 9
To evaluate the triple integral using spherical coordinates, we need to express the integrand and the volume element in terms of spherical coordinates.
Let's start by converting the given equations of the spheres to spherical coordinates.
In spherical coordinates, the position vector can be written as (r, θ, φ), where r is the radial distance from the origin, θ is the polar angle (measured from the positive z-axis), and φ is the azimuthal angle (measured from the positive x-axis).
For the equation x^2 + y^2 + z^2 = 49:
r^2 = x^2 + y^2 + z^2
r^2 = 49
In spherical coordinates, this becomes:
r = √49
r = 7
Similarly, for the equation x^2 + y^2 + z^2 = 81:
r = √81
r = 9
Now, let's express the integrand and the volume element in spherical coordinates.
The integrand:
e^-(x^2+y^2+z^2) / √(x^2+y^2+z^2)
can be rewritten in terms of r:
e^-(r^2) / √r^2
e^(-r^2) / r
The volume element:
dV = dx dy dz
In spherical coordinates, the volume element can be expressed as:
dV = r^2 sin(θ) dr dθ dφ
Now, we have all the necessary components to evaluate the triple integral in spherical coordinates:
∫∫∫ (e^(-r^2) / r) * r^2 sin(θ) dr dθ dφ
The limits for the triple integral are as follows:
r: 7 to 9
θ: 0 to π
φ: 0 to 2π
Therefore, the triple integral becomes:
∫[0 to 2π] ∫[0 to π] ∫[7 to 9] (e^(-r^2) / r) * r^2 sin(θ) dr dθ dφ
To evaluate this integral, you can integrate with respect to r first, then θ, and finally φ. The final result will be the evaluated value of the triple integral.