NOTES FOR CALC III STEWART 5TH

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NOTES FOR CALC IIISTEWART 5TH

• Section 17.6 PARAMETRIC SURFACES AND THEIR AREAS

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DEFINITION
A parametric surface is a set of three functions
or in vector form
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DEFINITION
If the parametric surface is a surface of
revolution (of y f(x) about the x-axis) then
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DEFINITION
If a parametric surface is given by
then its area, as (u, v) ranges over D, is
or, in the case that we can express z as z f(x,
y),
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EXAMPLE 1 4 pg 1142
Identify the surface
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EXAMPLE 1 4 pg 1142
Identify the surface
This is a surface of revolution where f(x) x is
rotated about y 0.
ANSWER A double cone about the x-axis
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EXAMPLE 1 4 pg 1142
Identify the surface
This is a surface of revolution where f(x) x is
rotated about y 0.
ANSWER A double cone about the x-axis
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EXAMPLE 2 22
Find a parametric representation of the part of
the sphere x2 y2 z2 16 that lies between
the planes z -2 and z 2.
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EXAMPLE 2 22
Find a parametric representation of the part of
the sphere x2 y2 z2 16 that lies between
the planes z -2 and z 2.
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EXAMPLE 2 22
Find a parametric representation of the part of
the sphere x2 y2 z2 16 that lies between
the planes z -2 and z 2.
Spherical would work well here (since we are
describing part of a sphere)
z
4
2
y
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EXAMPLE 2 22
Find a parametric representation of the part of
the sphere x2 y2 z2 16 that lies between
the planes z -2 and z 2.
Spherical would work well here (since we are
describing part of a sphere)
4
?
2
2
4
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EXAMPLE 2 22
Find a parametric representation of the part of
the sphere x2 y2 z2 16 that lies between
the planes z -2 and z 2.
Spherical would work well here (since we are
describing part of a sphere)
4
?
2
2
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EXAMPLE 2 22
Find a parametric representation of the part of
the sphere x2 y2 z2 16 that lies between
the planes z -2 and z 2.
Spherical would work well here (since we are
describing part of a sphere)
4
?
2
2
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EXAMPLE 2 22
Find a parametric representation of the part of
the sphere x2 y2 z2 16 that lies between
the planes z -2 and z 2.
Spherical would work well here (since we are
describing part of a sphere)
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EXAMPLE 3 28
Find parametric equations for the surface
obtained by rotating the curve x 4y2 y4, -2
y 2, about the y-axis.
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EXAMPLE 3 28
Find parametric equations for the surface
obtained by rotating the curve x 4y2 y4, -2
y 2, about the y-axis.
A surface of revolution of y f(x) about the
x-axis is
A surface of revolution of x f(y) about the
y-axis is
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EXAMPLE 3 28
Find parametric equations for the surface
obtained by rotating the curve x 4y2 y4, -2
y 2, about the y-axis.
A surface of revolution of y f(x) about the
x-axis is
ANSWER
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EXAMPLE 4 34
Find an equation of the tangent plane to the
parametric surface
r(u, v) uvi usin vj vcos uk at the point
u 0, v ?.
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EXAMPLE 4 34
Find an equation of the tangent plane to the
parametric surface
r(u, v) uvi usin vj vcos uk at the point
u 0, v ?.
ru(0, ?) ltv, sin v, -vsin ugt lt ?, 0, 0gt
rv(0, ?) ltu, ucos v, cos ugt lt0, 0, 1gt
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EXAMPLE 4 34
Find an equation of the tangent plane to the
parametric surface
r(u, v) uvi usin vj vcos uk at the point
u 0, v ?.
ru(0, ?) ltv, sin v, -vsin ugt lt ?, 0, 0gt
rv(0, ?) ltu, ucos v, cos ugt lt0, 0, 1gt
A normal would then be
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EXAMPLE 4 34
Find an equation of the tangent plane to the
parametric surface
r(u, v) uvi usin vj vcos uk at the point
u 0, v ?.
ru(0, ?) ltv, sin v, -vsin ugt lt ?, 0, 0gt
rv(0, ?) ltu, ucos v, cos ugt lt0, 0, 1gt
A normal would then be
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EXAMPLE 4 34
Find an equation of the tangent plane to the
parametric surface
r(u, v) uvi usin vj vcos uk at the point
u 0, v ?.
A normal is
A point of the plane is r(0, ?) 0i 0j ? k
-?y
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EXAMPLE 4 34
Find an equation of the tangent plane to the
parametric surface
r(u, v) uvi usin vj vcos uk at the point
u 0, v ?.
A normal is
A point of the plane is r(0, ?) 0i 0j ? k
n?X n?r
-?y 0
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EXAMPLE 4 34
Find an equation of the tangent plane to the
parametric surface
r(u, v) uvi usin vj vcos uk at the point
u 0, v ?.
A normal is
A point of the plane is r(0, ?) 0i 0j ? k
n?X n?r
-?y 0
ANSWER y 0
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EXAMPLE 5 38
Find the surface area of z 1 3x 2y2 that
lies above the triangle with vertices (0, 0), (0,
1) and (2, 1).
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EXAMPLE 5 38
Find the surface area of z 1 3x 2y2 that
lies above the triangle with vertices (0, 0), (0,
1) and (2, 1).
y
1
x 2y
zx 3 zy 4y
2 x
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EXAMPLE 5 38
Find the surface area of z 1 3x 2y2 that
lies above the triangle with vertices (0, 0), (0,
1) and (2, 1).
y
1
x 2y
zx 3 zy 4y
2 x
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EXAMPLE 5 38
Find the surface area of z 1 3x 2y2 that
lies above the triangle with vertices (0, 0), (0,
1) and (2, 1).
y
1
x 2y
zx 3 zy 4y
2 x
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EXAMPLE 5 38
Find the surface area of z 1 3x 2y2 that
lies above the triangle with vertices (0, 0), (0,
1) and (2, 1).
y
1
x 2y
zx 3 zy 4y
2 x
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EXAMPLE 5 38
Find the surface area of z 1 3x 2y2 that
lies above the triangle with vertices (0, 0), (0,
1) and (2, 1).
y
1
x 2y
zx 3 zy 4y
2 x
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ? , 0 u 1.
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ? , 0 u 1.
ru cosvi sin vj rv -usin vi ucos vj k
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ? , 0 u 1.
ru cosvi sin vj rv -usin vi ucos vj k
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ? , 0 u 1.
ru cosvi sin vj rv -usin vi ucos vj k
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ? , 0 u 1.
ru cosvi sin vj rv -usin vi ucos vj k
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ? , 0 u 1.
ru cosvi sin vj rv -usin vi ucos vj k
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ?, 0 u 1.
ru cosvi sin vj rv -usin vi ucos vj k
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ?, 0 u 1.
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ?, 0 u 1.
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ?, 0 u 1.
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EXAMPLE 5 44
Find the surface area of the helicoid r(u, v)
ucosvi usin vj vk, 0 v ?, 0 u 1.
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End of Section 17.6