MAT2615 Assignment 4 (QUALITY ANSWERS) 2025

This document contains workings, explanations and solutions to the MAT2615 Assignment 4 (QUALITY ANSWERS) 2025. For assistance whats-app us on 0.6.8..8.1.2..0.9.3.4... 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) (b) Sketch S and the XY-projection of S. (3) (c) Use a surface integral to evaluate the area of S. (8) [21] 2. (Chapter 17,Section 19.1) Consider the intersection R between the two circles x2 + y2 = 2 and (x − 2)2 + y2 = 2. y R x (a) Find a 2-dimensional vector field F = (M(x, y),N(x, y)) such that @N @x − @M @y = 1. (3) (b) Using this F and Green’s theorem (Theorem 19.1.1), write the area integral Z ZR 1 dA as a line integral. [Hint: Any function y = f (x) can be parametrised by r (t ) =