Question

Evaluate the triple integral ∭e^xy dv, where E is the solid tetrahedron.

a) 0
b) 1
c) e
d) [infinity]

Answer 1

To evaluate the triple integral ∭e^(xy) dv, we need to determine the limits of integration for each variable and convert the integral to cylindrical or spherical coordinates. The limits of integration will be 0 ≤ r ≤ 1 - z, 0 ≤ θ ≤ 2pi, and 0 ≤ z ≤ 1. Substituting these limits into the integral will allow us to evaluate it.

Step-by-step explanation:

To evaluate the triple integral ∬e^(xy) dv, we need to determine the limits of integration for each variable. Since the solid is a tetrahedron, we can consider the region bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 1.

Converting the integral to cylindrical or spherical coordinates may make it easier to evaluate. For cylindrical coordinates, we have x = rcos(theta), y = rsin(theta), and z = z. The limits of integration will be:

0 ≤ r ≤ 1 - z, 0 ≤ theta ≤ 2pi, and 0 ≤ z ≤ 1.

Substituting these limits into the integral and evaluating, we can find the value of the triple integral.