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)
(b) Sketch S and the XY-projection of S. (3)
(c) Use a surface integral to evaluate the area of S. (
[21]


𝑷𝒓𝒐𝒃𝒍𝒆𝒎 𝟏

(𝒂) 𝑪𝒐𝒎𝒑𝒖𝒕𝒆 𝒕𝒉𝒆 𝒗𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝑽𝑽 𝒖𝒔𝒊𝒏𝒈 𝒔𝒑𝒉𝒆𝒓𝒊𝒄𝒂𝒍 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔

𝑇ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑉𝑉 𝑖𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑎𝑏𝑜𝑣𝑒 𝑏𝑦 𝑡ℎ𝑒 ℎ𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒 𝑧 = 1 − 𝑥2 − 𝑦2𝑧
= 1 − 𝑥2 − 𝑦2 𝑎𝑛𝑑 𝑏𝑒𝑙𝑜𝑤 𝑏𝑦 𝑡ℎ𝑒 𝑐𝑜𝑛𝑒 𝑧 = 𝑥2 + 𝑦2 − 1𝑧
= 𝑥2 + 𝑦2 − 1. 𝑇𝑜 𝑐𝑜𝑚𝑝𝑢𝑡𝑒 𝑡ℎ𝑒 𝑣𝑜𝑙𝑢𝑚𝑒, 𝑤𝑒 𝑢𝑠𝑒 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠:

𝑥 = 𝜌𝑠𝑖𝑛 𝜙𝑐𝑜𝑠 𝜃, 𝑦 = 𝜌𝑠𝑖𝑛 𝜙𝑠𝑖𝑛 𝜃, 𝑧 = 𝜌𝑐𝑜𝑠 𝜙𝑥 = 𝜌𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝜃, 𝑦
= 𝜌𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜃, 𝑧 = 𝜌𝑐𝑜𝑠𝜙

𝑇ℎ𝑒 ℎ𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒 𝑧 = 1 − 𝑥2 − 𝑦2𝑧 = 1 − 𝑥2 − 𝑦2 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑡𝑜 𝜌 = 1𝜌
= 1, 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑛𝑒 𝑧 = 𝑥2 + 𝑦2 − 1𝑧 = 𝑥2 + 𝑦2 − 1 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑡𝑜 𝜙
= 𝜋4𝜙 = 4𝜋. 𝑇ℎ𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑖𝑠:

𝑑𝑉 = 𝜌2𝑠𝑖𝑛 𝜙 𝑑𝜌 𝑑𝜙 𝑑𝜃𝑑𝑉 = 𝜌2𝑠𝑖𝑛𝜙𝑑𝜌𝑑𝜙𝑑𝜃
𝑇ℎ𝑒 𝑙𝑖𝑚𝑖𝑡𝑠 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝑎𝑟𝑒:

 𝜌𝜌: 𝐹𝑟𝑜𝑚 00 𝑡𝑜 11,
 𝜙𝜙: 𝐹𝑟𝑜𝑚 00 𝑡𝑜 𝜋44𝜋,
 𝜃𝜃: 𝐹𝑟𝑜𝑚 00 𝑡𝑜 2𝜋2𝜋.
𝑇ℎ𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 𝑖𝑠: