# MATH 251: Calculus 3, SET8

## 16: Vector Calculus

### 16.6: Parametric Surfaces and Their Areas

These problems are done with the CAS. See Hand Solutions for details.

#### 1. [1120/6]

The surface below is a portion of an elliptic cylinder.
syms s t
fsurf(3*cos(t), s, sin(t), [-1 1 0 2*pi], 'MeshDensity', 12)
axis equal; axis([-3 3 -1 1 -1 1])
xticks(-3:3); yticks(-1:1); zticks(-1:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1120/6')

#### 2. [1120/12]

Behold the red pillow!
syms u v
fsurf(cos(u), sin(u)*sin(v), cos(v), 'r', [0 2*pi 0 2*pi], 'MeshDensity', 16)
axis equal; axis([-1 1 -1 1 -1 1])
xticks(-1:1); yticks(-1:1); zticks(-1:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1120/12')

#### 3. [1120/18]

A squished tin can is missing its lids.
syms u v
fsurf(sin(u), cos(u)*sin(v), sin(v), [0 2*pi 0 2*pi], 'MeshDensity', 16)
axis equal; axis([-1 1 -1 1 -1 1])
xticks(-1:1); yticks(-1:1); zticks(-1:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1120/18')

#### 4. [1120/24]

Here it the top half of a portion of a circular cylinder with axis of symmetry the y-axis.
syms t y
fsurf(3*cos(t), y, 3*sin(t), [0 pi -4 4], 'MeshDensity', 14)
axis equal; axis([-3 3 -4 4 -3 3])
xticks(-3:3:3); yticks(-4:4:4); zticks(-3:3:3)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1120/24')

#### 5. [1120/30]

A horn on its side is rendered.
syms t u
fsurf(cos(t)/u, u, sin(t)/u, [0 2*pi 1 5], 'MeshDensity', 14)
axis equal; axis([-1 1 1 5 -1 1])
xticks(-1:1); yticks(1:5); zticks(-1:1)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1120/30')

#### 6. [1121/36]

This is the tin can from #3. We find the tangent plane to the surface at a point.
syms u v x y z
r = [sin(u) cos(u)*sin(v) sin(v)]
r =
P = subs(r, [u v], [pi/6 pi/6])
P =
ru = diff(r,u), rv = diff(r,v)
ru =
rv =
ruP = subs(ru, [u v], [pi/6 pi/6]), rvP = subs(rv, [u v], [pi/6 pi/6])
ruP =
rvP =
n = cross(ruP,rvP)
n =
TP_P = dot(n,[x y z]) == dot(n,P)
TP_P =
TP_P = expand( TP_P * (-16/sqrt(3)) ) % equivalent, w/o fractions
TP_P =
TP_P_alt = z == solve(TP_P, z)
TP_P_alt =
%
figure
fsurf(rhs(TP_P_alt), [0 1 0 2], 'm', 'MeshDensity', 12)
axis([0 1 0 2 0 2]); xticks(0:1); yticks(0:2); zticks(0:2)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1121/36: Planes are flat!')

#### 7. [1121/42]

The surface area of a sliver of a cone is computed.
syms x y z
r = [x y sqrt(x^2 + y^2)]
r =
rx = diff(r,x), ry = diff(r,y)
rx =
ry =
n = cross(rx,ry)
n =
mag_n = simplify(sqrt(sum(n.^2)))
mag_n =
S = int(int(mag_n, y, x^2, x), x, 0, 1)
S =
S_appx = double(S) % in cm^2
S_appx = 0.2357
%
figure
[X Y Z] = hvsd(z==r(3), [y x^2 x], [x 0 1]) % voodoo magic
X = t
Y =
Z =
fsurf(X,Y,Z, [0 1 0 1], 'MeshDensity', 8)
axis equal; axis([0 1 0 1 0 1.5])
xticks(0:0.5:1); yticks(0:0.5:1); zticks(0:0.5:1.5)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1121/42: Slice of cone')

#### 8. [1121/48]

The surface area of a helicoid (spiral ramp) is computed.
syms u v
r = [u*cos(v) u*sin(v) v]
r =
ru = diff(r,u), rv = diff(r,v)
ru =
rv =
n = cross(ru,rv)
n =
mag_n = simplify(sqrt(sum(n.^2)))
mag_n =
S = int(int(mag_n, v, 0, pi), u, 0, 1)
S =
S_appx = double(S) % in cm^2
S_appx = 3.6059
%
figure
fsurf(r(1), r(2), r(3), [0 1 0 pi], 'MeshDensity', 12)
axis equal; axis([-1 1 0 1 0 3])
xticks(-1:1); yticks(0:1); zticks(0:3)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 1121/48: helicoid (spiral ramp)')

#### 9. [1121/54]

The surface area of a bizarre surface is approximated.
syms x y z
r = [x y (1+x^2)/(1+y^2)]
r =
rx = diff(r,x), ry = diff(r,y)
rx =
ry =
n = cross(rx,ry)
n =
mag_n = simplify(sqrt(sum(n.^2)))
mag_n =
S = int(int(mag_n, y, -(1-abs(x)), 1-abs(x)), x, -1, 1)
S =
S_appx = double(S) % in cm^2
S_appx = 2.6959
%
figure
[X Y Z] = hvsd(z==r(3), [y -(1-abs(x)), 1-abs(x)], [x -1 1]) % voodoo magic
X = t
Y =
Z =
fsurf(X,Y,Z, [-1 1 0 1], 'MeshDensity', 12)
axis equal; axis([-1 1 -1 1 0 2])
xticks(-1:1); yticks(-1:1); zticks(0:2)
xlabel('x'); ylabel('y'); zlabel('z')