, 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: 𝑽𝒐𝒍𝒖𝒎𝒆 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑼𝒔𝒊𝒏𝒈 𝑺𝒑𝒉𝒆𝒓𝒊𝒄𝒂𝒍 𝑪𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆
𝑮𝒊𝒗𝒆𝒏 𝑹𝒆𝒈𝒊𝒐𝒏 𝑽𝑽𝑽:
𝐵𝑜𝑢𝑛𝑑𝑒𝑑 𝑎𝑏𝑜𝑣𝑒 𝑏𝑦 𝑡ℎ𝑒 ℎ𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒: 𝑧 = 𝑝1 − 𝑥2 −
𝑦2𝑧 = 𝑝_1 − 𝑥^2 − 𝑦^2𝑧 = 𝑝1 − 𝑥2 − 𝑦2
𝐵𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑒𝑙𝑜𝑤 𝑏𝑦 𝑡ℎ𝑒 𝑐𝑜𝑛𝑒: 𝑧 = 𝑝(𝑥2 + 𝑦2) − 1𝑧 =
𝑝(𝑥^2 + 𝑦^2) − 1𝑧 = 𝑝(𝑥2 + 𝑦2) − 1
(𝒂) 𝑪𝒐𝒎𝒑𝒖𝒕𝒆 𝒕𝒉𝒆 𝒗𝒐𝒍𝒖𝒎𝒆 𝒖𝒔𝒊𝒏𝒈 𝒔𝒑𝒉𝒆𝒓𝒊𝒄𝒂𝒍 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔
1. 𝐶𝑜𝑛𝑣𝑒𝑟𝑡 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝑠 𝑡𝑜 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠:
o 𝑥 = 𝑟𝑐𝑜𝑠 𝜃, 𝑦 = 𝑟𝑠𝑖𝑛 𝜃, 𝑧 = 𝜌𝑐𝑜𝑠 𝜙𝑥 = 𝑟\𝑐𝑜𝑠\
𝑡ℎ𝑒𝑡𝑎,\𝑞𝑢𝑎𝑑 𝑦 = 𝑟\𝑠𝑖𝑛\𝑡ℎ𝑒𝑡𝑎,\𝑞𝑢𝑎𝑑 𝑧 =
\𝑟ℎ𝑜\𝑐𝑜𝑠\𝑝ℎ𝑖𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃, 𝑧 = 𝜌𝑐𝑜𝑠𝜙
o 𝐺𝑖𝑣𝑒𝑛 𝑧 = 𝑝1 − 𝑥2 − 𝑦2𝑧 = 𝑝_1 − 𝑥^2 −
𝑦^2𝑧 = 𝑝1 − 𝑥2 − 𝑦2, 𝑐𝑜𝑛𝑣𝑒𝑟𝑡 𝑡𝑜: 𝜌𝑐𝑜𝑠 𝜙 = 𝑝1 −
𝑟2\𝑟ℎ𝑜\𝑐𝑜𝑠\𝑝ℎ𝑖 = 𝑝_1 − 𝑟^2𝜌𝑐𝑜𝑠𝜙 = 𝑝1 − 𝑟2
o 𝐺𝑖𝑣𝑒𝑛 𝑧 = 𝑝(𝑥2 + 𝑦2) − 1𝑧 = 𝑝(𝑥^2 + 𝑦^2) −
1𝑧 = 𝑝(𝑥2 + 𝑦2) − 1, 𝑐𝑜𝑛𝑣𝑒𝑟𝑡 𝑡𝑜: 𝜌𝑐𝑜𝑠 𝜙 = 𝑝𝑟2 −
1\𝑟ℎ𝑜\𝑐𝑜𝑠\𝑝ℎ𝑖 = 𝑝 𝑟^2 − 1𝜌𝑐𝑜𝑠𝜙 = 𝑝𝑟2 − 1
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: 𝑽𝒐𝒍𝒖𝒎𝒆 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑼𝒔𝒊𝒏𝒈 𝑺𝒑𝒉𝒆𝒓𝒊𝒄𝒂𝒍 𝑪𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆
𝑮𝒊𝒗𝒆𝒏 𝑹𝒆𝒈𝒊𝒐𝒏 𝑽𝑽𝑽:
𝐵𝑜𝑢𝑛𝑑𝑒𝑑 𝑎𝑏𝑜𝑣𝑒 𝑏𝑦 𝑡ℎ𝑒 ℎ𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒: 𝑧 = 𝑝1 − 𝑥2 −
𝑦2𝑧 = 𝑝_1 − 𝑥^2 − 𝑦^2𝑧 = 𝑝1 − 𝑥2 − 𝑦2
𝐵𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑒𝑙𝑜𝑤 𝑏𝑦 𝑡ℎ𝑒 𝑐𝑜𝑛𝑒: 𝑧 = 𝑝(𝑥2 + 𝑦2) − 1𝑧 =
𝑝(𝑥^2 + 𝑦^2) − 1𝑧 = 𝑝(𝑥2 + 𝑦2) − 1
(𝒂) 𝑪𝒐𝒎𝒑𝒖𝒕𝒆 𝒕𝒉𝒆 𝒗𝒐𝒍𝒖𝒎𝒆 𝒖𝒔𝒊𝒏𝒈 𝒔𝒑𝒉𝒆𝒓𝒊𝒄𝒂𝒍 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔
1. 𝐶𝑜𝑛𝑣𝑒𝑟𝑡 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝑠 𝑡𝑜 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠:
o 𝑥 = 𝑟𝑐𝑜𝑠 𝜃, 𝑦 = 𝑟𝑠𝑖𝑛 𝜃, 𝑧 = 𝜌𝑐𝑜𝑠 𝜙𝑥 = 𝑟\𝑐𝑜𝑠\
𝑡ℎ𝑒𝑡𝑎,\𝑞𝑢𝑎𝑑 𝑦 = 𝑟\𝑠𝑖𝑛\𝑡ℎ𝑒𝑡𝑎,\𝑞𝑢𝑎𝑑 𝑧 =
\𝑟ℎ𝑜\𝑐𝑜𝑠\𝑝ℎ𝑖𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃, 𝑧 = 𝜌𝑐𝑜𝑠𝜙
o 𝐺𝑖𝑣𝑒𝑛 𝑧 = 𝑝1 − 𝑥2 − 𝑦2𝑧 = 𝑝_1 − 𝑥^2 −
𝑦^2𝑧 = 𝑝1 − 𝑥2 − 𝑦2, 𝑐𝑜𝑛𝑣𝑒𝑟𝑡 𝑡𝑜: 𝜌𝑐𝑜𝑠 𝜙 = 𝑝1 −
𝑟2\𝑟ℎ𝑜\𝑐𝑜𝑠\𝑝ℎ𝑖 = 𝑝_1 − 𝑟^2𝜌𝑐𝑜𝑠𝜙 = 𝑝1 − 𝑟2
o 𝐺𝑖𝑣𝑒𝑛 𝑧 = 𝑝(𝑥2 + 𝑦2) − 1𝑧 = 𝑝(𝑥^2 + 𝑦^2) −
1𝑧 = 𝑝(𝑥2 + 𝑦2) − 1, 𝑐𝑜𝑛𝑣𝑒𝑟𝑡 𝑡𝑜: 𝜌𝑐𝑜𝑠 𝜙 = 𝑝𝑟2 −
1\𝑟ℎ𝑜\𝑐𝑜𝑠\𝑝ℎ𝑖 = 𝑝 𝑟^2 − 1𝜌𝑐𝑜𝑠𝜙 = 𝑝𝑟2 − 1
