Question

cual es el área
2x²+y³ +4y​

Answer 1

Respuesta:

Alternate form:

2 x^2 + y (y^2 + 4)

Real root:

y≈0.48075 (1.73205 sqrt(27 x^4 + 64) - 9 x^2)^(1/3) - 2.77345/(1.73205 sqrt(27 x^4 + 64) - 9 x^2)^(1/3)

Roots:

y = (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3)/3^(2/3) - 4/(3^(1/3) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3))

y = (2 (1 + i sqrt(3)))/(3^(1/3) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3)) - ((1 - i sqrt(3)) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3))/(2 3^(2/3))

y = (2 (1 - i sqrt(3)))/(3^(1/3) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3)) - ((1 + i sqrt(3)) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3))/(2 3^(2/3))

Polynomial discriminant:

Δ_x = -8 (y^3 + 4 y)

Integer root:

x = 0, y = 0

Properties as a function:

Domain

R^2

Range

R (all real numbers)

Roots for the variable y:

y = (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3)/3^(2/3) - 4/(3^(1/3) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3))

y = (2 (1 + i sqrt(3)))/(3^(1/3) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3)) - ((1 - i sqrt(3)) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3))/(2 3^(2/3))

y = (2 (1 - i sqrt(3)))/(3^(1/3) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3)) - ((1 + i sqrt(3)) (sqrt(3) sqrt(27 x^4 + 64) - 9 x^2)^(1/3))/(2 3^(2/3))

Partial derivatives:

d/dx(2 x^2 + y^3 + 4 y) = 4 x

d/dy(2 x^2 + y^3 + 4 y) = 3 y^2 + 4

Indefinite integral:

integral(2 x^2 + 4 y + y^3) dx = (2 x^3)/3 + x y^3 + 4 x y + constant

Definite integral over a disk of radius R:

integral integral_(x^2 + y^2<R^2)(2 x^2 + y^3 + 4 y) dx dy = (π R^4)/2

Definite integral over a square of edge length 2 L:

integral_(-L)^L integral_(-L)^L (2 x^2 + 4 y + y^3) dy dx = (8 L^4)/3

Espero te sirva :)