Question

Use the cylindrical coordinates to evaluate ∬∬∬e^z dv, where E is the region bounded by the surfaces z = r, z = 0, and z = 1.

a) π
b) e
c) 1/e
d) e^2

Answer 1

To evaluate the given triple integral ∬∬∬e^z dv using cylindrical coordinates, substitute the limits and volume element into the integral. Simplify the integral by integrating with respect to r, then θ, and finally z. The value of the triple integral is π(e-1).

Step-by-step explanation:

To evaluate the given triple integral ∬∬∬e^z dv using cylindrical coordinates, we need to express the volume element dv in terms of the cylindrical coordinates. In cylindrical coordinates, dv = r dr dθ dz. The region E is bounded by the surfaces z = r, z = 0, and z = 1.

Since z = r lies on the surface, we can substitute r for z in the integral limits. Thus, the limits of integration for z are r from 0 to 1, for r are from 0 to 1, and for θ are from 0 to 2π. Substituting these limits and the corresponding volume element, the triple integral becomes:

∫∫∫e^z dv = ∫θ=02π ∫r=01 ∫z=r1 e^z (r dr dθ dz)

Simplifying the integral, we have ∫θ=02π ∫r=01 e^z r dr dθ. Integrating with respect to r first, we get ∫θ=02π [(e^z)/2] dθ. Integrating with respect to θ yields [πe^z]. Finally, integrating with respect to z from r to 1 gives [π(e-1)]. Therefore, the value of the triple integral ∬∬∬e^z dv is π(e-1).