Final answer:
To evaluate the triple integral in spherical coordinates, convert the integral into spherical coordinates and apply the radius bounds from √12 to √36, polar angle from 0 to π, and azimuthal angle from 0 to 2π, simplifying the integral for calculation.
Step-by-step explanation:
The student asked how to use spherical coordinates to evaluate the triple integral ∃∃∃ e^{-(x^2 + y^2 + z^2)/√x^2 + y^2 + z^2} dV, where E' is the region bounded by the spheres x^2 + y^2 + z^2 = 36 and x^2 + y^2 + z^2 = 12.
To solve this problem, we convert the Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), where r is the radius 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 in the xy-plane. In spherical coordinates, the volume element dV is given by r^2 sin(θ)drdθdφ. The integral becomes easier to evaluate as the function inside the integral depends only on r.
The limits for r are from √12 to √36, those for θ are from 0 to π, and for φ from 0 to 2π. The integral simplifies to an expression involving e^(-r) and the powers of r, which can be integrated using standard calculus techniques. Completing the integral over the specified limits will yield the value for the triple integral over the region E'.