MAT2615 Assignment 4 (COMPLETE ANSWERS) 2025

MAT2615 Assignment 4
(COMPLETE ANSWERS) 2025
100% GUARANTEEED

, MAT2615 Assignment 4 (COMPLETE
ANSWERS) 2025
1. (Section 14.6, Chapter 17) Let V be a region in
R3 bounded above by the hemisphere z = p1 − x2
��� y2 and below by the cone z = px2 + y2 − 1.
Let S be the surface of V (consisting of the
hemisphere on top and the paraboloid below). (a)
Compute the volume of V using spherical
coordinates. (10)
To compute the volume of the region V using spherical coordinates, we follow
these steps:
Step 1: Convert to Spherical Coordinates

Spherical coordinates are defined as:
x=ρ sin θ cos ϕ , y=ρ sin θ sin ϕ , z =ρ cos θ .

The volume element in spherical coordinates is:
2
dV =ρ sinθ dρdθ dϕ .

Step 2: Identify the Boundaries

The given region is bounded above by the hemisphere:
z= p1−x 2− y 2

and below by the cone:
2 2
z= p x + y −1.

In spherical coordinates:
 The hemisphere equation transforms as z= p1−r 2, or equivalently in spherical
coordinates ρ cos θ= p1−ρ2.
 The cone equation transforms to z= p r 2−1, or equivalently ρ cos θ= p ρ2−1.