14.3. Double Integrals in Polar Coordinates
Ex. 1
var('x, y, r, t')
f(x, y) = 3*x + 4*y^2
u = r*cos(t)
v = r*sin(t)
integral(integral(f(u, v)*r, r, 1, 2), t, 0, pi)
15/2*pi 15/2*pi |
Ex. 2
var('x, y, r, t')
f(x, y) = 1 - x^2 - y^2
u = r*cos(t)
v = r*sin(t)
integral(integral(f(u, v)*r, r, 0, 1), t, 0, 2*pi)
1/2*pi 1/2*pi |
Ex. 3
polar_plot(cos(2*t), (t, -pi/4, pi/4))
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var('r, t')
integral(integral(r, r, 0, cos(2*t)), t, -pi/4, pi/4)
1/8*pi 1/8*pi |
Ex. 4
var('x, y, z')
p1 = plot3d(x^2 + y^2, (x, 0, 2), (y, -1, 1), opacity = 0.4)
p2 = implicit_plot3d(x^2 + y^2 == 2*x, (x, 0, 2), (y, -1, 1), (z, 0, 5), color = 'green', opacity = 0.4)
p3 = plot3d(0, (x, 0, 2), (y, -1, 1), opacity = 0.4, color = 'orange')
p1 + p2 + p3
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Sleeping...
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var('x, y, r, t')
f(x, y) = x^2 + y^2
u = r*cos(t)
v = r*sin(t)
integral(integral(f(u, v)*r, r, 0, 2*cos(t)), t, -pi/2, pi/2)
3/2*pi 3/2*pi |
Exercise 1
var('x, y')
f1(x, y) = 3*x^2 + 3*y^2
f2(x, y) = 4 - x^2 - y^2
p1 = plot3d(f1, (x, -1, 1), (y, -1, 1), opacity = 0.4)
p2 = plot3d(f2, (x, -1, 1), (y, -1, 1), opacity = 0.4, color = 'orange')
show(p1 + p2, aspect_ratio = [1, 1, 1])
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Sleeping...
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var('r, t')
u = r*cos(t)
v = r*sin(t)
integral(integral((f2(u, v) - f1(u, v))*r, r, 0, 1), t, 0, 2*pi)
2*pi 2*pi |
14.5. Surface Area
Ex. 1
var('x, y')
f(x, y) = x^2 + 2*y
fx = diff(f(x, y), x)
fy = diff(f(x, y), y)
integral(integral(sqrt(1 + fx^2 + fy^2), y, 0, x), x, 0, 1)
-5/12*sqrt(5) + 9/4 -5/12*sqrt(5) + 9/4 |
Ex. 2
var('x, y, r, t')
f(x, y) = x^2 + y^2
fx = diff(f(x, y), x)
fy = diff(f(x, y), y)
u = r*cos(t)
v = r*sin(t)
integral(integral(sqrt(1 + fx^2 + fy^2).subs(x = u, y = v)*r, r, 0, 3), t, 0, 2*pi)
-1/6*pi + 1/6*(37*sqrt(37))*pi -1/6*pi + 1/6*(37*sqrt(37))*pi |
Exercise 1
var('x, y, z')
p1 = implicit_plot3d(x^2 + y^2 + z^2 == 4*z, (x, -2, 2), (y, -2, 2), (z, 0, 4), opacity = 0.3)
p2 = plot3d(x^2 + y^2, (x, -2, 2), (y, -2, 2), opacity = 0.3, color = 'red')
p1 + p2
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Sleeping...
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solve([x^2 + y^2 + z^2 == 4*z, x^2 + y^2 == z], x, y, z)
[[x == r1, y == -sqrt(-r1^2 + 3), z == 3], [x == r2, y == sqrt(-r2^2 + 3), z == 3], [x == r3, y == -I*r3, z == 0], [x == r4, y == I*r4, z == 0]] [[x == r1, y == -sqrt(-r1^2 + 3), z == 3], [x == r2, y == sqrt(-r2^2 + 3), z == 3], [x == r3, y == -I*r3, z == 0], [x == r4, y == I*r4, z == 0]] |
solve(x^2 + y^2 + z^2 == 4*z, z)
[z == -sqrt(-x^2 - y^2 + 4) + 2, z == sqrt(-x^2 - y^2 + 4) + 2] [z == -sqrt(-x^2 - y^2 + 4) + 2, z == sqrt(-x^2 - y^2 + 4) + 2] |
var('r, t')
f(x, y) = sqrt(-x^2 - y^2 + 4) + 2
fx = diff(f(x, y), x)
fy = diff(f(x, y), y)
u = r*cos(t)
v = r*sin(t)
integral(integral(sqrt(1 + fx^2 + fy^2).subs(x = u, y = v)*r, r, 0, sqrt(3)), t, 0, 2*pi)
4*pi 4*pi |
14.6. Triple Integrals
Ex. 1
var('x, y, z')
f(x, y, z) = x*y*z^2
integral(integral(integral(f(x, y, z), x, 0, 1), y, -1, 2), z, 0, 3)
27/4 27/4 |
Ex. 2
var('x, y, z')
f(x, y, z) = z
integral(integral(integral(f(x, y, z), z, 0, 1 - x - y), y, 0, 1 - x), x, 0, 1)
1/24 1/24 |
Ex. 3
var('x, y, z, r, t')
f(x, y, z) = sqrt(x^2 + z^2)
P(x, z) = integral(f(x, y, z), y, x^2 + z^2, 4)
u = r*cos(t)
v = r*sin(t)
integral(integral((P(u, v)*r).simplify_full(), r, 0, 2), t, 0, 2*pi)
128/15*pi 128/15*pi |
Ex. 5
var('x, y, z')
integral(integral(integral(1, z, 0, 2 - x - 2*y), y, x/2, 1 - x/2), x, 0, 1)
1/3 1/3 |
Exercise 1
var('x, y, z')
p1 = plot3d(x^2 - 1, (x, -2, 2), (y, -2, 2), opacity = 0.3)
p2 = plot3d(1 - x^2, (x, -2, 2), (y, -2, 2), opacity = 0.3, color = 'red')
p3 = implicit_plot3d(y == 0, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.3, color = 'green')
p4 = implicit_plot3d(y == 2, (x, -2, 2), (y, -2, 2), (z, -2, 2), opacity = 0.3, color = 'orange')
p1 + p2 + p3 + p4
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Sleeping...
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solve(x^2 - 1 == 1 - x^2, x)
[x == -1, x == 1] [x == -1, x == 1] |
integral(integral(integral(x - y, z, x^2 - 1, 1 - x^2), y, 0, 2), x, -1, 1)
-16/3 -16/3 |
Exercise 2
var('x, y, z')
p1 = implicit_plot3d(x^2 + z^ 2 == 4, (x, -2, 2), (y, -2, 6), (z, -4, 4), opacity = 0.3)
p2 = implicit_plot3d(y == -1, (x, -2, 2), (y, -2, 6), (z, -4, 4), opacity = 0.3, color = 'red')
p3 = implicit_plot3d(y + z == 4, (x, -2, 2), (y, -2, 6), (z, -4, 4), opacity = 0.3, color = 'green')
show(p1 + p2 + p3, aspect_ratio = [1, 1, 1])
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Sleeping...
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var('x, y, z, r, t')
P(x, z) = integral(1, y, -1, 4 - z)
u = r*cos(t)
v = r*sin(t)
integral(integral((P(u, v)*r).simplify_full(), r, 0, 2), t, 0, 2*pi)
20*pi 20*pi |
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