Para to Quad
True or False
The parametrization of a surface is unique
False
Parametrize the part of the cylinder x^2 + y^2 = 16 that lies above the xy-plane and below z = 3.
r(theta, z) = 4 cos theta i + 4 sin theta j + zk
0 <= theta <= 2pi
0 <= z <= 3
A surface S is defined by x^2 + z^2 = 25 and 0<= y <=10.
Find an equation of the tangent plane to S at the point (-3, 5, 4).
3x - 4z = -25
A smooth surface has a ________ at each point
a) normal plane
b) tangent plane
c) coordinate plane
b) tangent plane
Describe or draw the parametric surface using the equation.
r(u,v) = u sin v i + u cos v j + u k
0 <= u <= 4, 0 <= v <= 2pi
Upside down cone or funnel shape
Find the parametrization for the surface
The part of the plane z = 4 - x - 2y that lies in the first octant.
r(u,v) = ui + vj + (4- u - 2v)k
0 <= u <= 4, 0 <= v <= 2- (1/2)uA surface S is defined by x^2 + z^2 = 25 and 0<= y <=10.
Find an equation of the normal line to the tangent plane to S at the point (-3, 5, 4).
Normal line to the tangent plane contains the point (-3, 5, 4) and is parallel to 3i - 4k.
Parametrization of the normal line is r(t) = (-3 + 3t)i + 5j + (4-4t)k.
Find the surface area of the part of the surface z = f(x,y) = 2/3 (x^3/2 + y^3/2) that lies above the region R, a rectangle enclosed by the lines x = 0, x = 1, y = 0, and y = 2.
4/15 * (33 - 9sqrt3 - 4sqrt2)
Find a rectangular equation for the parametric surface
x(u,v) = u - 5v, y(u,v) = 2u, z(u,v) = -u + v + 1
x + 2y + 5z = 5
Find a parametrization for the surface.
The part of the surface z = sin(x^2*y) that lies above the region bounded by the graphs of y = x + 2 and y = x^2
r(u,v) = ui + vj + sin(u^2v)k
-1 <= u <= 2
u^2 <= v <= u + 2
Find the surface area of the part of the paraboloid z = f(x,y) = 1- x^2 - y^2 that lies above the xy-plane.
pi/6 * (5sqrt5 - 1)
Find the rectangular equation for the parametric surface.
x(u,v) = u cos v, y(u,v) = u sin v, z(u,v) = u
0 <= u <= 2, 0 <= v <= pi
0 <= z <= 2
Part of the paraboloid z = 9 - x^2 - y^2 lies inside the cylinder (x-1)^2 + y^2 = 1.
a) Parametrize the surface using rectangular coordinates
b) Parametrize the surface using cylindrical coordinates
a) r(u,v) = ui + vj + (9 - u^2 - v^2)k
-sqrt(1 - (u-1)^2) <= v <= sqrt(1 - (u-1)^2), 0 <= u <= 2
b) r(r, theta) = (1 + r cos theta) i + r sin theta j + (8 - r^2 - 2r cos theta)k
0 <= r <= 1, 0 <= theta <= 2pi
a) Find the tangent plane to the torus parametrized by r(u,v) = cos u (3 + cos v)i + sin u (3 + cos v)j + sinvk at the point ((3 sqrt(2))/2) , ((3 sqrt(2))/2), 1).
b) Find an equation of the normal line to the tangent plane at the same point.
a) z = 1
b) r(t) = ((3 sqrt(2))/2) i + ((3 sqrt(2))/2) j + (1 + 3t) k
Find the surface area of S
S is parametrized by r(u,v)= ucosvi + u^3j + usinvk.
0<= u <= 1, -pi <= v <= pi
(1/6)pi * (3sqrt(10) + ln(3 + sqrt(10)))
r(u,v) = a cosh u i + (b sinh u cos v) j + (c sinh u sin v) k
A torus is formed by rotating a circle about another circle.
Parametrize the torus obtained by rotating a circle of radius a > 0 about a circle of radius b > 0.
r(u,v) = (b + a cos u) cos vi + (b + a cos u) sin vj + a sin uk
Find an equation of the tangent plane to Dini's surface at the point (pi/3, pi/4).
Dini's surface is defined by the parametric equations:
x = 6 cos u sin v, y = 6 sin u sin v, z = 6[cos v + ln(tan v/2)] + u
0.01 <= v <= 1, 0 <= u <= 6pi
(9 - 3(sqrt(6))/2) (x - 3(sqrt(2))/2) + (9 sqrt(3) + 3sqrt(2)/2 ) (y - (3 sqrt(6) / 2)) - 18 {z - (pi/3 + 6[sqrt(2)/2 +ln(tan pi/8)])} = 0
Find the surface area S of the helicoid parametrized by r(u,v) = ucos(2v)i + usin(2v)j + vk.
0 <= u <= 1, 0 <= v <= 2pi
sqrt(5)*pi + (1/2)pi * ln(2 + sqrt(5))