# MATH 251: Calculus 3, SET8

## 15: Multiple Integrals

### 15.5: Surface Area

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

#### 1. [1028/2]

The surface area of the part of a plane lying inside a circular cylinder is calculated.
Here the double integral of over the circular disk of radius 5 centered at the origin is that multiple of the area of the disk.
syms r theta x y z
plane = 6*x + 4*y + 2*z == 1, f = solve(plane, z)
plane =
f =
grad_f = gradient(f, [x y]), g = sqrt(1 + norm(grad_f)^2)
g =
S = g*pi*5^2, S_appx = double(S) % in cm^2
S =
S_appx = 293.8691
%
figure
x = r*cos(theta), y = r*sin(theta), z = 1/2 - 2*y - 3*x
x =
y =
z =
fsurf(x, y, z, [0 5 0 2*pi], 'm', 'MeshDensity', 14)
xlabel('x'); ylabel('y'); zlabel('z'); axis equal
title('SET8, 1028/2')

#### 2. [1028/4]

syms x y positive; syms z % The triangular shadow region is in the first quadrant.
surf_eq = 2*y + 4*z - x^2 == 5, f = solve(surf_eq, z)
surf_eq =
f =
grad_f = gradient(f, [x y]), g = sqrt(1 + norm(grad_f)^2)
g =
S = int(int(g, y, 0, 2*x), x, 0, 2), S_appx = double(S) % in cm^2
S =
S_appx = 5.2732
%
figure
[X Y Z] = hvsd(z==f, [y 0 2*x], [x 0 2]); % voodoo transformation
fsurf(X,Y,Z, [0 2 0 1], 'EdgeColor', 'none'); hold on % the surface
fsurf(X,Y,sym(0), [0 2 0 1], 'y', 'EdgeColor', 'none') % shadow region
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; xticks(0:2); yticks(0:4); zticks(0:2)
title('SET8, 1028/4'); view(-66,20)

#### 3. [1028/6]

syms x y positive
f = sqrt(4-x^2), grad_f = gradient(f, [x y])
f =
h = simplify(1 + sum(grad_f.^2))
h =
g = 2/sqrt(4-x^2) % g = sqrt(1 + norm(grad_f)^2)
g =
S = int(int(g, x, 0, 1), y, 0, 1), S_appx = double(S) % in cm^2
S =
S_appx = 1.0472
%
figure
fsurf(f, [0 1 0 1], 'c', 'MeshDensity', 8); hold on
fsurf(sym(0), [0 1 0 1], 'y', 'EdgeColor', 'none')
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; xticks(0:1); yticks(0:1); zticks(0:2)
title('SET8, 1028/6')

#### 4. [1028/8]

syms x y positive
f = 2/3*(x^(3/2) + y^(3/2)), grad_f = gradient(f, [x y])
f =
g = sqrt(1 + norm(grad_f)^2)
g =
S = int(int(g, x, 0, 1), y, 0, 1), S_appx = double(S) % in cm^2
S =
S_appx = 1.4066
%
figure
fsurf(f, [0 1 0 1], 'MeshDensity', 10); hold on
fsurf(sym(0), [0 1 0 1], 'y', 'EdgeColor', 'none')
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([0 1 0 1 0 1.5])
xticks(0:1); yticks(0:1); zticks(0.0:0.5:1.5)
title('SET8, 1028/8')

#### 5. [1028/10]

syms phi r theta x y
f = sqrt(4 - x^2 - y^2), grad_f = gradient(f, [x y])
f =
h = simplify(1 + sum(grad_f.^2))
h =
g = 2 / sqrt(4-r^2)
g =
S = int(int(g*r, r, 0, sqrt(3)), theta, 0, 2*pi), S_appx = double(S) % in cm^2
S =
S_appx = 12.5664
%
figure
fsurf(2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi), ...
[0 pi 0 2*pi], 'g', 'MeshDensity', 16); hold on
fsurf(sym(1), [-3 3 -3 3], 'r', 'MeshDensity', 8)
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([-3 3 -3 3 -2 2])
xticks(-2:2:2); yticks(-2:2:2); zticks(-2:2:2)
title('SET8, 1028/10')

#### 6. [1028/12]

syms phi r theta x y
% sph = x^2 + y^2 + z^2 - 4*z + 4 == 4
% sph = x^2 + y^2 + (z-2)^2 = 4
f = 2 + sqrt(4-x^2-y^2) % f = 2 + sqrt(4-4^2)
f =
h = simplify(1 + sum(grad_f.^2))
h =
g = 2 / sqrt(4-r^2)
g =
S = int(int(g*r, r, 0, sqrt(3)), theta, 0, 2*pi), S_appx = double(S) % in cm^2
S =
S_appx = 12.5664
%
figure
fsurf(2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2+2*cos(phi), ...
[0 pi 0 2*pi], 'g', 'MeshDensity', 16); hold on
fsurf(r*cos(theta), r*sin(theta), r^2, [0 2 0 2*pi], 'r', 'MeshDensity', 16)
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([-2 2 -2 2 0 4])
xticks(-2:2:2); yticks(-2:2:2); zticks(0:2:4)
title('SET8, 1028/12'); alpha 0.5

#### 7. [1028/14]

syms r theta x y
f = cos(x^2 + y^2)
f =
h = simplify(1 + sum(grad_f.^2))
h =
g = sqrt( simplify( subs(h, [x y], [r*cos(theta) r*sin(theta)]) ) )
g =
S = int(int(g*r, r, 0, 1), theta, 0, 2*pi), S_appx = double(S) % in cm^2
S =
S_appx = 4.1073
%
figure
fsurf(r*cos(theta), r*sin(theta), cos(r^2), [0 1 0 2*pi], 'MeshDensity', 14); hold on
fsurf(r*cos(theta), r*sin(theta), sym(0), [0 1 0 2*pi], 'm', 'EdgeColor', 'none')
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; axis([-1 1 -1 1 0 1])
xticks(-1:1); yticks(-1:1); zticks(0:1)
title('SET8, 1028/14')

#### 8. [1029/16]

syms x y
f = x*y + x^2 + y^2
f =
h = expand(1 + sum(grad_f.^2))
h =
g = sqrt(h)
g =
S = int(int(g, x, 0, 2), y, 0, 2), S_appx = double(S) % in cm^2
S =
S_appx = 17.7165
%
figure
fsurf(f, [0 2 0 2], 'MeshDensity', 14); hold on
fsurf(sym(0), [0 2 0 2], 'm', 'EdgeColor', 'none')
xlabel('x'); ylabel('y'); zlabel('z'); axis equal
xticks(0:2); yticks(0:2); zticks(0:6:12)
title('SET8, 1028/16')

#### 9. [1029/18]

syms x y
f = 1 + x + y + x^2
f =
h = expand(1 + sum(grad_f.^2))
h =
g = sqrt(h)
g =
S = int(int(g, x, -2, 1), y, -1, 1), S_appx = double(S) % in cm^2
S =
S_appx = 12.9431
%
figure
fsurf(f, [-2 1 -1 1], 'MeshDensity', 14); hold on
xlabel('x'); ylabel('y'); zlabel('z'); axis equal
xticks(-2:1); yticks(-1:1); % zticks(0:6:12)
title('SET8, 1028/18')

#### 10. [1029/20]

syms x y z
f = (1+x^2) / (1+y^2)
f =
h = expand(1 + sum(grad_f.^2))
h =
g = sqrt(h)
g =
S = int(int(g, y, -(1-abs(x)), 1-abs(x)), x, -1, 1), S_appx = double(S) % in cm^2
S =
S_appx = 2.6959
%
figure
[X Y Z] = hvsd(z==f, [y -(1-abs(x)) 1-abs(x)], [x -1 1]); % voodoo transformation
fsurf(X,Y,Z, [-1 1 0 1], 'MeshDensity', 14); hold on % the surface
fsurf(X,Y,sym(0), [-1 1 0 1], 'm', 'EdgeColor', 'none') % shadow region
xlabel('x'); ylabel('y'); zlabel('z')
axis equal; xticks(-1:1); yticks(-1:1); zticks(0:2)
title('SET8, 1028/20'); view(28,28)