Question

use cylindrical coordinates. evaluate 2(x3 xy2) dv, where e is the solid in the first octant that lies beneath the paraboloid z = 4 − x2 − y2. e

Answer 1

To evaluate the integral using cylindrical coordinates, we first convert the variables and volume element to cylindrical coordinates and then determine the integration limits for the given solid. The volume element dv becomes r dz dr d theta, and the limits include r from 0 to 2, theta from 0 to pi/2, and z from 0 to 4 - r^2.

Step-by-step explanation:

To evaluate the integral 2(x3 xy2) dv using cylindrical coordinates for the solid E in the first octant and beneath the paraboloid z = 4 − x2 − y2, we must convert the volume element and the function inside the integral to cylindrical coordinates. The volume element in cylindrical coordinates dv becomes rdzdrdθ, and the function 2(x3 xy2) becomes 2(r3 cos3(θ) r3 sin2(θ) cos(θ)) after converting x and y to r cos(θ) and r sin(θ) respectively.

The limits for r go from 0 to 2, as the paraboloid intersects the plane z = 0 when x2 + y2 = 4 (hence r = 2), and the angle θ ranges from 0 to 2pi/(pi/2) to cover the first octant. For z, it ranges from 0 to 4 − r2 to stay beneath the paraboloid. After setting up the integral with these limits, we can evaluate it by successive integration.