Let S, be the part of the paraboloid z = æ² + y² that is inside the cylinder a? + y² = 1,S2 the part of the cylinder a? + y? = 1 between the planes z = 0 and z = 1 and Sy the part of the plane z = 0 that is inside the cylinder a?+ y? = 1. Let Rbe the region that is enclosed by S1, Są and Sg. a) Assume that units of a, y, z are metres. Calculate the surface area of Si. b) Using polar coordinates, calculate the volume of R. Consider the vector field v(x) = (4x + 3y)i +(4y – 62)j + (7x – 2)k . c) Calculate f(u•n)dS, where n is pointing away from the z-axis.
Let S, be the part of the paraboloid z = æ² + y² that is inside the cylinder a? + y² = 1,S2 the part of the cylinder a? + y? = 1 between the planes z = 0 and z = 1 and Sy the part of the plane z = 0 that is inside the cylinder a?+ y? = 1. Let Rbe the region that is enclosed by S1, Są and Sg. a) Assume that units of a, y, z are metres. Calculate the surface area of Si. b) Using polar coordinates, calculate the volume of R. Consider the vector field v(x) = (4x + 3y)i +(4y – 62)j + (7x – 2)k . c) Calculate f(u•n)dS, where n is pointing away from the z-axis.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Do all parts in detail
Transcribed Image Text:Let S, be the part of the paraboloid z = 2? + y? that is inside the cylinder ? + y? = 1, S2, the part of
the cylinder æ? + y? = 1 between the planes z = 0 and z = 1 and Sg the part of the plane z = 0 that is
inside the cylinder a?+ y? = 1. Let Rbe the region that is enclosed by S1, Sz and Sg.
a)
Assume that units of z, y, z are metres. Calculate the surface area of Sı.
b)
Using polar coordinates, calculate the volume of R.
Consider the vector field v(æ) = (4x + 3y)i + (4y – 62)j+(7x – 2)k -
c)
Calculate SS(v •n)dS, where n is pointing away from the z-axis.
d)
Show that fS(v•n)dS = 0. Hint:you don't need any parametrisation for this.
Calculate SS(v• n)dS, where n is pointing away from the z-axis, onlyusing the results from (b),(c),(d)
and the Gauss Theorem.
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