T4: Funciones · Cálculo de áreas — MATEMATICAS II PEvAU Andaluc%2525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525C3%2525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525252525ADa 2024

Cálculo de áreas

Problema

2024 · Ordinaria · Reserva

Considera la función $f : \mathbb{R} \to \mathbb{R}$ definida por $f(x) = x^3 - 6x^2 + 8x$ .

a) Calcula los puntos de corte de la gráfica de

f

con los ejes de coordenadas y esboza dicha gráfica.b) Calcula la suma de las áreas de los recintos acotados y limitados por la gráfica de

f

y el eje de abscisas.

Puntos de corteÁrea bajo la curvaEsbozo de gráfica

a) Calcula los puntos de corte de la gráfica de

f

con los ejes de coordenadas y esboza dicha gráfica.

Para hallar los puntos de corte con el eje de abscisas (eje $X$ ), igualamos la función a cero:

x^3 - 6x^2 + 8x = 0

Factorizamos la expresión extrayendo factor común $x$ :

x(x^2 - 6x + 8) = 0

Resolvemos la ecuación de segundo grado $x^2 - 6x + 8 = 0$ :

x = \frac{6 \pm \sqrt{36 - 32}}{2} = \frac{6 \pm 2}{2} \implies x_1 = 4, \, x_2 = 2

Los puntos de corte con el eje $X$ son $(0, 0)$ , $(2, 0)$ y $(4, 0)$ . El punto de corte con el eje $Y$ se obtiene evaluando $f(0) = 0$ , que es el punto $(0, 0)$ ya calculado.Para el esbozo, observamos que al ser una función polinómica de tercer grado con coeficiente principal positivo, el límite cuando $x \to \infty$ es $+\infty$ y cuando $x \to -\infty$ es $-\infty$ . Presenta un máximo relativo entre $x=0$ y $x=2$ , y un mínimo relativo entre $x=2$ y $x=4$ .

b) Calcula la suma de las áreas de los recintos acotados y limitados por la gráfica de

f

y el eje de abscisas.

Los recintos están delimitados por los puntos de corte con el eje $X$ : el primero en el intervalo $[0, 2]$ y el segundo en $[2, 4]$ . El área total $A$ es la suma de las integrales definidas en valor absoluto:

A = \int_{0}^{2} (x^3 - 6x^2 + 8x) \, dx + \left| \int_{2}^{4} (x^3 - 6x^2 + 8x) \, dx \right|

Calculamos primero la primitiva de la función:

F(x) = \int (x^3 - 6x^2 + 8x) \, dx = \frac{x^4}{4} - 2x^3 + 4x^2

Evaluamos en el primer recinto $[0, 2]$ :

I_1 = \left[ \frac{x^4}{4} - 2x^3 + 4x^2 \right]_{0}^{2} = (4 - 16 + 16) - 0 = 4

Evaluamos en el segundo recinto $[2, 4]$ :

I_2 = \left[ \frac{x^4}{4} - 2x^3 + 4x^2 \right]_{2}^{4} = (64 - 128 + 64) - (4 - 16 + 16) = 0 - 4 = -4

Sumamos los valores absolutos para obtener el área total:

A = |4| + |-4| = 4 + 4 = 8 \text{ unidades de \'area}