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Integral definida: Área bajo la curva

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Integral definida: Área bajo la curva

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Quiz sobre área bajo la curva

Created by

Jaime Urzua
Jaime Urzua
Mexico

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Integral definida: Área bajo la curva
 

Integral definida: Área bajo la curvaOnline version

Quiz sobre área bajo la curva

by Jaime Urzua
1

Qué representa la integral definida ∫_a^b f(x) dx en geometría?

2

Si f(x) ≥ 0 en [a,b], ¿qué dice la integral ∫_a^b f(x) dx sobre el área?

3

¿Qué propiedad guía la integral si se cambia el eje de coordenadas con x = t + c?

4

¿Qué condición se necesita para que ∫_a^b f(x) dx represente una área positiva?

5

Si f(x) ≥ 0 y f(x) es continua, la integral ∫_a^b f(x) dx es:

6

¿Qué significa calcular ∫_2^5 (3x) dx?

7

Si ∫_a^b f(x) dx = 0, ¿qué puede ocurrir?

8

Para f(x) positiva en [a,b], ¿cómo cambia ∫_a^b f(x) dx si se reduce el intervalo a [a,c] con a≤c≤b?

9

¿Qué regla de cálculo se aplica para ∫_a^b [f(x)+g(x)] dx?

10

¿Qué significa el teorema fundamental del cálculo en relación a ∫_a^b f(x) dx?

Explicación

Representa el área entre la curva y el eje x en [a,b]. Las otras opciones describen conceptos distintos (volumen, suma, promedio).

Con f≥0, la integral da el área positiva. No es volumen, pendiente ni derivada.

Mover x por una constante cambia límites y conserva el valor numérico de la integral.

Solo cuando la función es no negativa se interpreta como área. Si f≤0, sería área en negativo.

Con continuidad y f≥0, la integral da un número finito que es el área.

Es la definición de área para la recta lineal 3x entre 2 y 5.

La única forma de que el área sea 0 es que f esté 0 casi everywhere en [a,b] o que se cancelen áreas positivas y negativas.

Reducir el intervalo disminuye o mantiene el valor; no puede aumentar si f≥0.

Linealidad de la integral: ∫(f+g)=∫f+∫g.

El teorema conecta la integral con una antiderivada F: ∫_a^b f = F(b)-F(a).

 
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