The Definite Integral
Understanding the Definite Integral
The definite integral represents the area under a curve between two points, providing a powerful tool for calculating areas, volumes, work, and many other quantities in calculus.
For a continuous function $f(x)$ on the interval $[a, b]$, the definite integral is defined as:
$$\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$$
where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a point in the $i$-th subinterval.
The definite integral has several key properties:
- Additivity over intervals: $\int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx = \int_{a}^{b} f(x) \, dx$
- Reverse limits: $\int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx$
- Constant multiple: $\int_{a}^{b} c \cdot f(x) \, dx = c \cdot \int_{a}^{b} f(x) \, dx$
- Sum of functions: $\int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx$
The definite integral is closely related to the antiderivative through the Fundamental Theorem of Calculus, which we'll explore in another section.
Notes and Resources
Key Takeaways
- The definite integral represents the net area under a curve between two points
- It can be calculated as the limit of Riemann sums or using the Fundamental Theorem of Calculus
- Numerical methods like the trapezoidal rule and Simpson's rule provide practical approximations
- Definite integrals have numerous applications in geometry, physics, engineering, and other fields
- The properties of definite integrals (additivity, linearity) make them powerful computational tools