def test_volumeintegrals():
cube = ParametricRegion((x, y, z), (x, 0, 1), (y, 0, 1), (z, 0, 1))
assert ParametricIntegral(1, cube) == 1
solidsphere = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\
(r, 0, 2), (theta, 0, 2*pi), (phi, 0, pi))
assert ParametricIntegral(C.x**2 + C.y**2, solidsphere) == -256 * pi / 15
region_under_plane1 = ParametricRegion((x, y, z), (x, 0, 3), (y, 0, -2*x/3 + 2),\
(z, 0, 6 - 2*x - 3*y))
region_under_plane2 = ParametricRegion((x, y, z), (x, 0, 3), (z, 0, 6 - 2*x - 3*y),\
(y, 0, -2*x/3 + 2))
assert ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane1) == \
ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane2)
assert ParametricIntegral(2 * C.x, region_under_plane2) == -9
def test_vector_integrate():
halfdisc = ParametricRegion((r * cos(theta), r * sin(theta)), (r, -2, 2),
(theta, 0, pi))
assert vector_integrate(C.x**2, halfdisc) == 4 * pi
vector_integrate(C.x, ParametricRegion(
(t, t**2), (t, 2, 3))) == -17 * sqrt(17) / 12 + 37 * sqrt(37) / 12
assert vector_integrate(C.y**3 * C.z, (C.x, 0, 3),
(C.y, -1, 4)) == 765 * C.z / 4
s1 = Segment(Point(0, 0), Point(0, 1))
assert vector_integrate(-15 * C.y, s1) == S(-15) / 2
s2 = Segment(Point(4, 3, 9), Point(1, 1, 7))
assert vector_integrate(C.y * C.i, s2) == -6
curve = Curve((sin(t), cos(t)), (t, 0, 2))
assert vector_integrate(5 * C.z, curve) == 10 * C.z
c1 = Circle(Point(2, 3), 6)
assert vector_integrate(C.x * C.y, c1) == 72 * pi
c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
assert vector_integrate(1, c2) == c2.circumference
triangle = Polygon((0, 0), (1, 0), (1, 1))
assert vector_integrate(C.x * C.i - 14 * C.y * C.j, triangle) == 0
p1, p2, p3, p4 = [(0, 0), (1, 0), (5, 1), (0, 1)]
poly = Polygon(p1, p2, p3, p4)
assert vector_integrate(-23 * C.z,
poly) == -161 * C.z - 23 * sqrt(17) * C.z
point = Point(2, 3)
assert vector_integrate(C.i * C.y - C.z, point) == ParametricIntegral(
C.y * C.i, ParametricRegion((2, 3)))
c3 = ImplicitRegion((x, y), x**2 + y**2 - 4)
assert vector_integrate(45, c3) == 360 * pi
c4 = ImplicitRegion((x, y), (x - 3)**2 + (y - 4)**2 - 9)
assert vector_integrate(1, c4) == 12 * pi
pl = Plane(Point(1, 1, 1), Point(2, 3, 4), Point(2, 2, 2))
raises(ValueError, lambda: vector_integrate(C.x * C.z * C.i + C.k, pl))
def test_parametric_surfaceintegrals():
semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\
(theta, 0, 2*pi), (phi, 0, pi/2))
assert ParametricIntegral(C.z, semisphere) == 8*pi
cylinder = ParametricRegion((sqrt(3)*cos(theta), sqrt(3)*sin(theta), z), (z, 0, 6), (theta, 0, 2*pi))
assert ParametricIntegral(C.y, cylinder) == 0
cone = ParametricRegion((v*cos(u), v*sin(u), v), (u, 0, 2*pi), (v, 0, 1))
assert ParametricIntegral(C.x*C.i + C.y*C.j + C.z**4*C.k, cone) == pi/3
triangle1 = ParametricRegion((x, y), (x, 0, 2), (y, 0, 10 - 5*x))
triangle2 = ParametricRegion((x, y), (y, 0, 10 - 5*x), (x, 0, 2))
assert ParametricIntegral(-15.6*C.y*C.k, triangle1) == ParametricIntegral(-15.6*C.y*C.k, triangle2)
assert ParametricIntegral(C.z, triangle1) == 10*C.z
def test_parametric_lineintegrals():
halfcircle = ParametricRegion((4*cos(theta), 4*sin(theta)), (theta, -pi/2, pi/2))
assert ParametricIntegral(C.x*C.y**4, halfcircle) == S(8192)/5
curve = ParametricRegion((t, t**2, t**3), (t, 0, 1))
field1 = 8*C.x**2*C.y*C.z*C.i + 5*C.z*C.j - 4*C.x*C.y*C.k
assert ParametricIntegral(field1, curve) == 1
line = ParametricRegion((4*t - 1, 2 - 2*t, t), (t, 0, 1))
assert ParametricIntegral(C.x*C.z*C.i - C.y*C.z*C.k, line) == 3
assert ParametricIntegral(4*C.x**3, ParametricRegion((1, t), (t, 0, 2))) == 8
helix = ParametricRegion((cos(t), sin(t), 3*t), (t, 0, 4*pi))
assert ParametricIntegral(C.x*C.y*C.z, helix) == -3*sqrt(10)*pi
field2 = C.y*C.i + C.z*C.j + C.z*C.k
assert ParametricIntegral(field2, ParametricRegion((cos(t), sin(t), t**2), (t, 0, pi))) == -5*pi/2 + pi**4/2