4.3
Integral definida
Considere una función real definida en un intervalo cerrado. Uno de los objetivos fundamentales del cálculo es determinar el área bajo la gráfica de d…
Consideremos una función sobre un intervalo cerrado. Para aproximar el área bajo la curva, dividir el intervalo en subintervalos iguales, formando rectángulos de igual anchura.
Selecciona un punto arbitrario dentro del subintervalo y utiliza el valor de la función ahí como la altura del rectángulo. Multiplicar la altura por el ancho da el área de ese rectángulo. El área total bajo la curva se aproxima sumando las áreas de todos los rectángulos y se denomina suma de Riemann.
La integral definida de una función sobre un intervalo dado se define como el límite de su suma de Riemann a medida que el número de subintervalos tiende al infinito y el ancho de cada rectángulo se vuelve infinitesimalmente pequeño, siempre que este límite exista.
Por ejemplo, imaginemos un coche que viaja a una velocidad variable. Divide el tiempo total de viaje en intervalos iguales.
En cada intervalo, selecciona un punto temporal arbitrario y toma la velocidad correspondiente. Multiplica la velocidad por la anchura del intervalo para estimar la distancia recorrida.
La suma de estos forma una suma de Riemann, y en este ejemplo, el área bajo la curva se interpreta físicamente como la distancia recorrida por el coche. Cuando la velocidad sigue siendo positiva, la distancia total se determina integrando la función de velocidad dentro de los límites.
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Q1: How does dividing an interval into subintervals help approximate area under a curve?
Dividing a closed interval into equal subintervals creates rectangles with uniform width. By selecting an arbitrary point within each subinterval and using the function value there as the rectangle's height, you can calculate each rectangle's area by multiplying height by width. Summing all rectangle areas produces a Riemann sum, which approximates the total area under the curve.
Q2: What is the relationship between Riemann sums and definite integrals?
A definite integral is defined as the limit of Riemann sums as the number of subintervals approaches infinity and each rectangle's width becomes infinitesimally small. This limiting process refines the approximation into an exact value representing the area under the curve over a specified interval, provided the limit exists.
Q3: How can you use integration to find the distance traveled by a moving object?
When a car travels with varying velocity, divide the total travel time into equal intervals and estimate distance in each interval by multiplying velocity at a chosen moment by the interval width. Summing these distances creates a Riemann sum. When velocity remains positive, integrating the velocity function over the time interval yields the exact total distance traveled.
Q4: Why does accuracy improve when you increase the number of subintervals?
As the number of subintervals increases, their widths decrease, making each rectangle's height more representative of the function's actual behavior over that interval. This finer partition reflects the curve's shape more closely, reducing the gap between the approximation and the true area, thereby improving accuracy.
Q5: What does it mean for a rectangle's height to be determined by an arbitrary point within a subinterval?
Within each subinterval, you can choose any point and evaluate the function at that location. The resulting function value becomes the rectangle's height for that subinterval. This flexibility in point selection is fundamental to constructing Riemann sums and demonstrates how different choices can yield different approximations of the area.
Q6: How does the definite integral represent a limit process?
The definite integral emerges when the partition of an interval becomes infinitely fine. As the number of rectangles grows without bound and each rectangle's width approaches zero, the Riemann sum converges to a limiting value. This limiting value represents the exact area under the curve and defines the definite integral.
Q7: What physical interpretation does the area under a velocity curve represent?
The area under a velocity-time curve represents the total distance traveled during that time interval. When velocity remains positive throughout, integrating the velocity function over the specified time limits yields the exact distance covered. This connection between integration and motion demonstrates how calculus models real-world phenomena.