# MATH 251: Calculus 3, SET8

## 12: Vectors and the Geometry of Space

### 12.6: Cylinders and Quadric Surfaces

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

#### 1. [839/4]

An elliptical cylinder is graphed.
syms theta z
figure
fsurf(cos(theta), 2*sin(theta), z, [0 2*pi -3 3], 'MeshDensity', 12)
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 839/4')

#### 2. [839/8]

An sinusoidal cylinder is plotted.
syms x y
figure
fsurf(x, y, sin(y), [-5 5 0 4*pi], 'MeshDensity', 20)
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 839/8')

#### 3. [840/28]

Here is a plot of a hyperbolic paraboloid of one sheet.
syms x z
eq = y == x^2 - z^2
eq =
figure
fsurf(x, x^2 - z^2, z, [-3 3 -3 3], 'MeshDensity', 20)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 840/28')
view(-26,26)

#### 4. [840/36]

Complete squares to see that the surface is a circular cone.
syms r x y z theta
eq1 = x^2 - y^2 - z^2 - 4*x - 2*z + 3 == 0
eq1 =
eq2 = x^2 - 4*x + 4 == y^2 + z^2 + 2*z + 1 - 3 + 4 - 1
eq2 =
eq3 = (x-2)^2 == y^2 + (z+1)^2
eq3 =
figure
fsurf(2+r, r*cos(theta), -1+r*sin(theta), [-3 3 0 2*pi], 'MeshDensity', 16)
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 840/36')

#### 5. [840/38]

Complete squares to see that the surface is an ellipsoid.
syms x y z phi theta
eq1 = 4*x^2 + y^2 + z^2 - 24*x - 8*y + 4*z + 55 == 0
eq1 =
eq2 = 4*(x^2 - 6*x + 9) + y^2 - 8*y + 16 + z^2 + 4*z + 4 == -55 + 36 + 16 + 4
eq2 =
eq3 = (x-3)^2 / (1/2)^2 + (y-4)^2 / 1^2 + (z+2)^2 / 1^2 == 1
eq3 =
figure
fsurf(3 + 1/2*sin(phi)*cos(theta), 4 + sin(phi)*sin(theta), -2 + cos(phi), ...
'm', [0 pi 0 2*pi], 'MeshDensity', 20)
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 840/38')

#### 6. [840/42]

Completing squares yields an elliptic paraboloid.
syms r t x y z
eq1 = x^2 - 6*x + 4*y^2 - z == 0
eq1 =
eq2 = z == x^2 - 6*x + 9 + 4*y^2 - 9
eq2 =
eq3 = z == (x-3)^2 + (y-0)^2 / (1/2)^2 - 9
eq3 =
figure
fsurf(3+r*cos(t), 1/2*r*sin(t), r^2 - 9, [0 3 0 2*pi], 'c', 'MeshDensity', 20)
axis equal; xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 840/42')

#### 7. [840/44]

Here is the region enclosed by two paraboloids. (They intersect at level .)
syms r t x y z
paraboloid1 = z == x^2 + y^2
paraboloid1 =
paraboloid2 = z == 2 - x^2 - y^2
paraboloid2 =
figure
fsurf(r*cos(t), r*sin(t), r^2, [0 1 0 2*pi], 'r', 'MeshDensity', 20); hold on
fsurf(r*cos(t), r*sin(t), 2 - r^2, [0 1 0 2*pi], 'g', 'MeshDensity', 20)
axis equal; xticks(-1:1); yticks(-1:1); zticks(0:2)
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 840/44')

#### 8. [841/46]

Take the line in the yz-plane and rotate it about the z-axis to produce a surface of revolution: a circular cone.
syms r t
figure
fsurf(r*cos(t), r*sin(t), 2*r, [-4 4 0 2*pi], 'MeshDensity', 20)
axis equal;
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 840/46')

#### 9. [841/48]

The geometrical description yields another circular cone.
syms r t x y z
eq1 = sqrt(y^2 + z^2) == 2*abs(x)
eq1 =
eq2 = x^2 == 1/4 * y^2 + 1/4 * z^2
eq2 =
figure
fsurf(r, 2*r*cos(t), 2*r*sin(t), [-4 4 0 2*pi], 'MeshDensity', 20)
axis equal;
xlabel('x'); ylabel('y'); zlabel('z')
title('SET8, 840/48')