2 x y^2 + 6 y^3 + 4 y + 18
Step-by-step explanation:
2×2 y + 8×3 - x y^2 - 2×3 + 3 x y^2 + 6 y^3
Result:
2 x y^2 + 6 y^3 + 4 y + 18
3D plot:
3D plot
Contour plot:
Contour plot
Alternate forms:
2 (x y^2 + 3 y^3 + 2 y + 9)
2 (y (y (x + 3 y) + 2) + 9)
y (y (2 x + 6 y) + 4) + 18
Real roots:
x≈-9.24225, y≈-0.797732
x≈-9.24225, y≈1.93924
Roots:
y = (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)/(9 2^(1/3)) - (2^(1/3) (18 - x^2))/(9 (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9
y = -((1 - i sqrt(3)) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3))/(18 2^(1/3)) + ((1 + i sqrt(3)) (18 - x^2))/(9 2^(2/3) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9
y = -((1 + i sqrt(3)) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3))/(18 2^(1/3)) + ((1 - i sqrt(3)) (18 - x^2))/(9 2^(2/3) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9
Polynomial discriminant:
Δ_y = -16 (36 x^3 - 4 x^2 - 972 x + 19779)
Integer roots:
x = -14, y = 1
x = -4, y = -1
Properties as a function:
Domain
R^2
Range
R (all real numbers)
Roots for the variable y:
y = (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)/(9 2^(1/3)) - (2^(1/3) (18 - x^2))/(9 (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9
y = -((1 - i sqrt(3)) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3))/(18 2^(1/3)) + ((1 + i sqrt(3)) (18 - x^2))/(9 2^(2/3) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9
y = -((1 + i sqrt(3)) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3))/(18 2^(1/3)) + ((1 - i sqrt(3)) (18 - x^2))/(9 2^(2/3) (-2 x^3 + 9 sqrt(3) sqrt(36 x^3 - 4 x^2 - 972 x + 19779) + 54 x - 2187)^(1/3)) - x/9
Partial derivatives:
d/dx(2 x y^2 + 6 y^3 + 4 y + 18) = 2 y^2
d/dy(2 x y^2 + 6 y^3 + 4 y + 18) = 4 x y + 18 y^2 + 4
Indefinite integral:
integral(18 + 4 y + 2 x y^2 + 6 y^3) dx = x^2 y^2 + 6 x y^3 + 4 x y + 18 x + constant
Definite integral over a disk of radius R:
integral integral_(x^2 + y^2<R^2)(2 x y^2 + 6 y^3 + 4 y + 18) dx dy = 18 π R^2
Definite integral over a square of edge length 2 L:
integral_(-L)^L integral_(-L)^L (18 + 4 y + 2 x y^2 + 6 y^3) dy dx = 72 L^2
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