MATH 251: Calculus 3, SET8

15: Multiple Integrals

15.4: Applications of Double Integrals

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

1. [1024/2]

The total electric charge on a circular disk is computed.
syms r theta
Q = int(int(r*r, r, 0, 1), theta, 0, 2*pi)
Q =
Q_appx = double(Q) % total charge in coulombs
Q_appx = 2.0944

2. [1025/6]

The mass and center of mass of a lamina is determined for a triangular region.
syms x y
L1 = y == 2*x, L2 = x + 2*y == 1
L1 =
L2 =
P = solve([L1 L2], [x y]); P = [P.x P.y]
P =
delta = x % density
delta = x
m = int(int(delta, x, y/2, 1-2*y), y, 0, 2/5)
m =
CM = 1/m * int(int(delta*[x y], x, y/2, 1-2*y), y, 0, 2/5)
CM =
m_appx = double(m) % g
m_appx = 0.0800
CM_appx = double(CM) % components in cm
CM_appx = 1×2
0.5167 0.1167
%
figure
X = [0 1 1/5 0]; Y = [0 0 2/5 0];
plot([-0.2 1.2], [0 0], 'k'); grid on; hold on
plot([0 0], [-0.1 1.5], 'k')
fill(X,Y,'y')
plot(X,Y, 'b', 'LineWidth', 2)
plot(31/60, 7/60, 'm+', 'MarkerSize', 14)
xlabel('x'); ylabel('y')
axis equal; axis([-0.2 1.2 -0.1 0.5])
xticks(0.0:0.2:1.0); yticks(0.0:0.2:0.4)
title('SET8, 1025/6')

3. [1025/10]

Same as #2 for a lamina bounded by a trigonometric function and the x-axis.
syms x y
delta = y % density
delta = y
m = int(int(delta, y, 0, cos(x)), x, -pi/2, pi/2)
m =
CM = 1/m * int(int(delta*[x y], y, 0, cos(x)), x, -pi/2, pi/2)
CM =
m_appx = double(m) % g
m_appx = 0.7854
CM_appx = double(CM) % components in cm
CM_appx = 1×2
0 0.5659
%
figure
t = linspace(-pi/2,pi/2); xt = t; yt = cos(t);
X = [xt -pi/2]; Y = [yt 0];
plot([-2 2], [0 0], 'k'); grid on; hold on
plot([0 0], [-0.5 1.5], 'k')
fill(X,Y,'y')
plot(X,Y, 'b', 'LineWidth', 2)
plot(0, 0.5659, 'm+', 'MarkerSize', 14)
xlabel('x'); ylabel('y')
axis equal; axis([-2 2 -0.5 1.5])
xticks(-1.5:0.5:1.5); yticks(-0.5:0.5:1.5)
title('SET8, 1025/10')

4. [1025/12]

Same as #2 for the quarter-circular unit disk in the first quadrant.
syms k r theta x y
delta = k*r^2 % density
delta =
x = r*cos(theta), y = r*sin(theta)
x =
y =
m = int(int(delta * r, r, 0, 1), theta, 0, pi/2)
m =
CM = 1/m * int(int(delta*[x y] * r, r, 0, 1), theta, 0, pi/2)
CM =
CM_appx = double(CM) % components in cm
CM_appx = 1×2
0.5093 0.5093
%
figure
t = linspace(0,pi/2); xt = cos(t); yt = sin(t);
X = [xt 0 1]; Y = [yt 0 0];
plot([-0.5 1.5], [0 0], 'k'); grid on; hold on
plot([0 0], [-0.5 1.5], 'k')
fill(X,Y,'y')
plot(X,Y, 'b', 'LineWidth', 2)
plot(0.5093, 0.5093, 'm+', 'MarkerSize', 14)
xlabel('x'); ylabel('y')
axis equal; axis([-0.5 1.5 -0.5 1.5])
xticks(-0.5:0.5:1.5); yticks(-0.5:0.5:1.5)
title('SET8, 1025/12')

5. [1025/14]

Same as #2 for semi-circular ring in the upper half-plane.
syms k r theta x y
delta = k/r % density
delta =
x = r*cos(theta), y = r*sin(theta)
x =
y =
m = int(int(delta * r, r, 1, 2), theta, 0, pi)
m =
CM = 1/m * int(int(delta*[x y] * r, r, 1, 2), theta, 0, pi)
CM =
CM_appx = double(CM) % components in cm
CM_appx = 1×2
0 0.9549
%
figure
t = linspace(0,pi); xt = cos(t); yt = sin(t);
X = [xt fliplr(2*xt) 1]; Y = [yt fliplr(2*yt) 0];
plot([-3 3], [0 0], 'k'); grid on; hold on
plot([0 0], [-1 0.8], 'k')
plot([0 0], [2 3], 'k')
fill(X,Y,'y')
plot(X,Y, 'b', 'LineWidth', 2)
plot(0, 3/pi, 'm+', 'MarkerSize', 14)
xlabel('x'); ylabel('y')
axis equal; axis([-3 3 -1 3])
xticks(-3:3); yticks(-1:3)
title('SET8, 1025/14')

6. [1025/18]

For the region in #2 (q.v.), various moments of inertia are found.
syms x y
delta = x % density
delta = x
Ix = int(int(delta*y^2, x, y/2, 1-2*y), y, 0, 2/5)
Ix =
Iy = int(int(delta*x^2, x, y/2, 1-2*y), y, 0, 2/5)
Iy =
I0 = Ix + Iy
I0 =
Ix_appx = double(Ix)
Ix_appx = 0.0017
Iy_appx = double(Iy)
Iy_appx = 0.0250
I0_appx = double(I0)
I0_appx = 0.0267

7. [1025/24]

Moments of inertia and radii of gyration are found for a lamina bounded by a trigonometric function and the x-axis.
syms k x y
delta = k % density
delta = k
m = int(int(delta, y, 0, sin(x)), x, 0, pi)
m =
Ix = int(int(delta*y^2, y, 0, sin(x)), x, 0, pi)
Ix =
Iy = int(int(delta*x^2, y, 0, sin(x)), x, 0, pi)
Iy =
x_bar_bar = simplify(sqrt(Iy/m))
x_bar_bar =
y_bar_bar = sqrt(Ix/m)
y_bar_bar =
x_bar_bar_appx = double(x_bar_bar)
x_bar_bar_appx = 1.7131
y_bar_bar_appx = double(y_bar_bar)
y_bar_bar_appx = 0.4714

8. [1025/26]

Mass, center of mass, and moments of inertia of a lamina are determined.
syms x y
delta = x^2*y^2 % density
delta =
m = int(int(delta, y, 0, x*exp(-x)), x, 0, 2)
m =
CM = 1/m * int(int(delta*[x y], y, 0, x*exp(-x)), x, 0, 2)
CM =
Ix = int(int(delta*y^2, y, 0, x*exp(-x)), x, 0, 2)
Ix =
Iy = int(int(delta*x^2, y, 0, x*exp(-x)), x, 0, 2)
Iy =
I0 = Ix + Iy
I0 =
m_appx = double(m) % g
m_appx = 0.0304
CM_appx = double(CM) % components in cm
CM_appx = 1×2
1.4205 0.2480
Ix_appx = double(Ix)
Ix_appx = 0.0020
Iy_appx = double(Iy)
Iy_appx = 0.0656
I0_appx = double(I0)
I0_appx = 0.0676

9. [1025/28]

We verify that a function is a joint density function (JDF), then compute two probabilities and two expected values.
syms x y
f(x,y) = 4*x*y % for 0 <= x <= 1, 0 <= y <= 1, and zero elsewhere
f(x, y) =
% (a)
JDF = int(int(f(x,y), x, 0, 1), y, 0, 1) % Integral over xy-plane is 1; f is a JDF.
JDF = 1
% (b)
b1 = int(int(f(x,y), x, 1/2, 1), y, 0, 1)
b1 =
b2 = int(int(f(x,y), x, 1/2, 1), y, 0, 1/2)
b2 =
% (c)
EX = int(int(x*f(x,y), x, 0, 1), y, 0, 1) % expected value of X
EX =
EY = int(int(y*f(x,y), x, 0, 1), y, 0, 1) % expected value of Y
EY =

10. [1025/29]

Same as #9 with a different joint density function.
syms x y
f(x,y) = 0.1*exp(-(0.5*x + 0.2*y)) % for x >= 0, y >= 0, and zero elsewhere
f(x, y) =
% (a)
JDF = int(int(f(x,y), x, 0, inf), y, 0, inf) % Integral over xy-plane is 1; f is a JDF.
JDF = 1
% (b)
b1 = int(int(f(x,y), x, 0, inf), y, 1, inf), b1_appx = double(b1)
b1 =
b1_appx = 0.8187
b2 = expand(int(int(f(x,y), x, 0, 2), y, 0, 4)), b2_appx = double(b2)
b2 =
b2_appx = 0.3481
% (c)
EX = int(int(x*f(x,y), x, 0, inf), y, 0, inf) % expected value of X
EX = 2
EY = int(int(y*f(x,y), x, 0, inf), y, 0, inf) % expected value of Y
EY = 5