MATH 251: Calculus 3, SET8

15: Multiple Integrals

15.3: Double Integrals in Polar Coordinates

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

1. [1014/4]

The region depicted in the figure for the statement of the problem is part of a circular disk of radius 3 centered at the origin. So we would definitely want to use polar coordinates to set up the following integral.

2. [1014/8]

syms r theta
f = simplify(2*r*cos(theta) - r*sin(theta))
f = 
I = int(int(f*r, r, 0, 2), theta, pi/4, pi/2)
I = 
I_appx = double(I)
I_appx = -0.3235
%
figure
t = linspace(pi/4,pi/2); xt = 2*cos(t); yt = 2*sin(t);
X = [xt 0 xt(1)]; Y = [yt 0 yt(1)];
fill(X,Y,'y'); grid on; hold on
plot(X,Y, 'LineWidth', 3)
plot([-1 3], [0 0], 'k')
plot([0 0], [-1 3], 'k')
xlabel('x'); ylabel('y')
title('SET8, 1014/8')
axis equal; axis([-1 2 -1 3])
xticks(0:2); yticks(0:2)

3. [1015/12]

syms r theta
f = cos(r)
f = 
I = int(int(f*r, r, 0, 2), theta, 0, 2*pi)
I = 
I_appx = double(I)
I_appx = 2.5287

4. [1015/16]

Via symmetry, the area is 4 times that in the first quadrant.
syms r t theta
A = 4*int(int(1*r, r, 0, 1-cos(theta)), theta, 0, pi/2)
A = 
A_appx = double(A) % in cm^2
A_appx = 0.7124
%
t = linspace(0,2*pi); r1 = 1 - cos(t); r2 = 1 + cos(t);
figure
polarplot(t, r1, 'b', 'LineWidth', 2); hold on
polarplot(t, r2, 'r')
title('SET8, 1015/16')

5. [1015/20]

syms r theta
V = int(int(r*r, r, 1, 2), theta, 0, 2*pi)
V_appx = double(V) % in cm^3
%
figure
fsurf(r*cos(theta), r*sin(theta), sym(0), [1 2 0 2*pi], 'r', 'EdgeColor', 'none'); hold on
fsurf(r*cos(theta), r*sin(theta), r, [1 2 0 2*pi], 'MeshDensity', 10)
axis equal; axis([-2 2 -2 2 0 2])
xticks(-2:2); yticks(-2:2); zticks(0:2)
xlabel('x'); ylabel('y')
title('SET8, 1015/20')
V = 
V_appx = 14.6608

6. [1015/24]

syms r theta
s3 = sqrt(3);
V = int(int((7 - (1 + 2*r^2))*r, r, 0, s3), theta, 0, pi/2)
V = 
V_appx = double(V) % in cm^3
V_appx = 7.0686
%
figure
fsurf(r*cos(theta), r*sin(theta), sym(7), [0 s3 0 pi/2], 'r', 'MeshDensity', 10); hold on
fsurf(r*cos(theta), r*sin(theta), 1+2*r^2, [0 s3 0 pi/2], 'MeshDensity', 10)
fsurf(r*cos(theta), r*sin(theta), sym(0), [0 s3 0 pi/2], 'm', 'MeshDensity', 10)
axis equal; axis([-3 3 -3 3 0 8])
xticks(-3:3:3); yticks(-3:3:3); zticks(0:2:8)
xlabel('x'); ylabel('y')
title('SET8, 1015/24')
view(130,23)

7. [1015/30]

syms r theta x y
syms a positive
s = sqrt(a^2 - y^2)
s = 
IR = int(int(2*x + y, x, -s, s), y, 0, a)
IR = 
IP = int(int((2*r*cos(theta) + r*sin(theta)) * r, r, 0, a), theta, 0, pi)
IP = 
%
figure
t = linspace(0,pi); xt = 2*cos(t); yt = 2*sin(t); % for example, a = 2
X = [xt 2]; Y = [yt 0];
fill(X,Y,'y'); grid on; hold on
plot(X,Y,'b', 'LineWidth', 3)
plot([-3 3], [0 0], 'k')
plot([0 0], [-1 3], 'k')
axis equal; xlabel('x'); ylabel('y')
axis([-3 3 -1 3]); xticks([-5 5]); yticks([-5 5])
text(-0.25, 2.25, 'a', 'FontSize', 14)
title('SET8, 1015/30')

8. [1015/32]

syms r theta x y
eq1 = y^2 == 2*x - x^2
eq1 = 
eq2 = lhs(eq1) - rhs(eq1) == 0
eq2 = 
circle = (x-1)^2 + (y-0)^2 == 1
circle = 
circ_p = subs(circle, [x y], [r*cos(theta) r*sin(theta)])
circ_p = 
circ_p_simp = simplify(circ_p) % Offset circle is r = 2*cos(theta).
circ_p_simp = 
I = int(int(r*r, r, 0, 2*cos(theta)), theta, 0, pi/2)
I = 
I_appx = double(I)
I_appx = 1.7778
%
figure
t = linspace(0,pi/2); rt = 2*cos(t); xt = rt.*cos(t); yt = rt.*sin(t);
X = [xt 0 2]; Y = [yt 0 0];
fill(X,Y,'y'); grid on; hold on
plot(X,Y,'b', 'LineWidth', 3)
plot([-1 3], [0 0], 'k')
plot([0 0], [-1 2], 'k')
axis equal; xlabel('x'); ylabel('y')
axis([-1 3 -1 2]); xticks(-1:3); yticks(-1:2)
title('SET8, 1015/32')

9. [1015/36]

For a field sprinkled with water, the total amount of water per hour and average amount of water per hour per square foot are computed.
syms r theta
total = expand ( int(int(exp(-r)*r, r, 0, 100), theta, 0, 2*pi) )
total = 
total_appx = double(total) % essentially 2*pi, in ft^3
total_appx = 6.2832
A = sym(pi*100^2)
A = 
avg_amt = total / A
avg_amt = 
avg_amt_appx = double(avg_amt) % in ft^3 per hour per ft^2
avg_amt_appx = 2.0000e-04

10. [1015/39]

syms r theta x y
x = r*cos(theta), y = r*sin(theta), s2 = sqrt(2);
x = 
y = 
I = int(int(x*y * r, r, 1, 2), theta, 0, pi/4)
I = 
I_appx = double(I)
I_appx = 0.9375
%
figure
t = linspace(0,pi/4); xt = cos(t); yt = sin(t);
X = [xt fliplr(2*xt) 1]; Y = [yt fliplr(2*yt), 0];
fill(X,Y,'y'); grid on; hold on
plot(X,Y,'b', 'LineWidth', 3)
plot([-1 3], [0 0], 'k')
plot([0 0], [-1 2], 'k')
plot([1 1], [0 1], 'b--', 'LineWidth', 1)
plot([s2,s2], [0 s2], 'b--', 'LineWidth', 1)
axis equal; xlabel('x'); ylabel('y')
axis([-0.5 2.5 -0.5 2]); xticks([-1:3]); yticks([-1:2])
title('SET8, 1015/39')