13) Evaluate the double integral ♫♫ cos(x² + y²) dy dx by rewriting in a different coordinate system. Sketch the region first before doing anything else. 14) Evaluate the double integral ſſxy dA, where R is the region in the first quadrant enclosed by y = √x, y = 6x, and y = 0. Π 15) Evaluate the triple integral √ √cose) √² r sin(e) dz dr do. What is the coordinate system being used in this integral? 16) Use a triple integral to show that the volume of a sphere with radius r ≥ 0 is given by the formula V = 1 ½πr³. 17) Use the Cross-Partial Test to show that the vector field F = (2xy³, 1 + 3x²y²) is a conservative vector field. 18) Find the potential function for each conservative vector field: (a) F(x, y) = (x, y) (b) G(x, y) = (cos(y) + y cos(x), sin(x) − x sin(y)) 19) Evaluate the line integral with respect to s along the parametric curve C: x = t, y = t², z = ²½ t³ ↓ 3x²yz ds C 20) Use the Divergence Theorem to evaluate the surface integral ſſ F · dS, where F = (3x+y, z, 5zx) and S is the boundary of the region between the paraboloid z = 4 − x² − y² and the xy-plane. (Hint: You will need to use a change of variables at some point.)
13) Evaluate the double integral ♫♫ cos(x² + y²) dy dx by rewriting in a different coordinate system. Sketch the region first before doing anything else. 14) Evaluate the double integral ſſxy dA, where R is the region in the first quadrant enclosed by y = √x, y = 6x, and y = 0. Π 15) Evaluate the triple integral √ √cose) √² r sin(e) dz dr do. What is the coordinate system being used in this integral? 16) Use a triple integral to show that the volume of a sphere with radius r ≥ 0 is given by the formula V = 1 ½πr³. 17) Use the Cross-Partial Test to show that the vector field F = (2xy³, 1 + 3x²y²) is a conservative vector field. 18) Find the potential function for each conservative vector field: (a) F(x, y) = (x, y) (b) G(x, y) = (cos(y) + y cos(x), sin(x) − x sin(y)) 19) Evaluate the line integral with respect to s along the parametric curve C: x = t, y = t², z = ²½ t³ ↓ 3x²yz ds C 20) Use the Divergence Theorem to evaluate the surface integral ſſ F · dS, where F = (3x+y, z, 5zx) and S is the boundary of the region between the paraboloid z = 4 − x² − y² and the xy-plane. (Hint: You will need to use a change of variables at some point.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
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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Do only question 15
Transcribed Image Text:13) Evaluate the double integral ♫♫
cos(x² + y²) dy dx by rewriting in a different coordinate system.
Sketch the region first before doing anything else.
14) Evaluate the double integral ſſxy dA, where R is the region in the first quadrant enclosed by y = √x, y =
6x, and y = 0.
Π
15) Evaluate the triple integral √ √cose) √² r sin(e) dz dr do. What is the coordinate system being used in this
integral?
16) Use a triple integral to show that the volume of a sphere with radius r ≥ 0 is given by the formula V = 1 ½πr³.
17) Use the Cross-Partial Test to show that the vector field F = (2xy³, 1 + 3x²y²) is a conservative vector field.
18) Find the potential function for each conservative vector field:
(a) F(x, y) = (x, y)
(b) G(x, y) = (cos(y) + y cos(x), sin(x) − x sin(y))
19) Evaluate the line integral with respect to s along the parametric curve C: x = t, y = t², z = ²½ t³
↓ 3x²yz ds
C
20) Use the Divergence Theorem to evaluate the surface integral ſſ F · dS, where F = (3x+y, z, 5zx) and S
is the boundary of the region between the paraboloid z = 4 − x² − y² and the xy-plane. (Hint: You will need to
use a change of variables at some point.)
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