Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 9 - x2 - y2. SOLUTION If we put z = 0 in the equation of the paraboloid, we get x2 + y2 = 9, so the solid lies under the paraboloid and above the circular disk D given by x2 + y2 s 9. In polar coordinates D is given by 0 srs 3,0 ses 2n. Since 9 - x2 - y2 = 9 -2, the volume is V= x² - y²)dA Video Example ) (9 - 23r dr de (9r - P)dr de If we had used rectangular coordinates instead of polar coordinates, then we would have obtained (9 - x? - y?)dA = (9 – 2 - y) dydx which is not easy to evaluate because it involves finding S(9 - x2,3/2dx
Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 9 - x2 - y2. SOLUTION If we put z = 0 in the equation of the paraboloid, we get x2 + y2 = 9, so the solid lies under the paraboloid and above the circular disk D given by x2 + y2 s 9. In polar coordinates D is given by 0 srs 3,0 ses 2n. Since 9 - x2 - y2 = 9 -2, the volume is V= x² - y²)dA Video Example ) (9 - 23r dr de (9r - P)dr de If we had used rectangular coordinates instead of polar coordinates, then we would have obtained (9 - x? - y?)dA = (9 – 2 - y) dydx which is not easy to evaluate because it involves finding S(9 - x2,3/2dx
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 9 -.
x² - y².
SOLUTION
If we put z = O in the equation of the paraboloid, we get x² + y2 = 9, so the solid lies under the paraboloid and above the circular disk D
given by x2 + y² < 9. In polar coordinates D is given by 0 <r < 3, 0 < 0 s 2n. Since 9 - x2 - y² = 9 - r2, the volume is
V =
(9 - x2 - y²)dA
Video Example )
r27
(9 - r2)r dr d0
d0
(9r -
= 2n
If we had used rectangular coordinates instead of polar coordinates, then we would have obtained
-3
V =
(9 – x² – y²) dydr
(9 -
which is not easy to evaluate because it involves finding S(9 - x²)3/2dx
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