Up until now, the integral has been our tool for measuring the space between a single curve and the static ground of the x-axis. But what if the floor itself is moving? In this lesson, we transcend the axis and learn to compute the area of regions trapped between two independent functional boundaries, $f(x)$ and $g(x)$.
The Geometry of Differences
To find the area $A$ of a region $S$ bounded by $y = f(x)$ and $y = g(x)$ between $x = a$ and $x = b$, we utilize the same Riemann Sum logic that built the foundations of calculus.
The Riemann Extension
We divide the region into $n$ vertical strips. If $x_i^*$ is a sample point in the $i$-th interval, the height of the approximating rectangle is not just $f(x_i^*)$, but the difference between the heights of the top and bottom curves:
$$h = f(x_i^*) - g(x_i^*)$$
Summation to Integration
As we increase the number of strips to infinity ($n \to \infty$), the sum of these rectangular areas converges to the definite integral:
核心公式:
$$A = \lim_{n \to \infty} \sum_{i=1}^{n} [f(x_i^*) - g(x_i^*)] \Delta x = \int_a^b [f(x) - g(x)] dx$$
where $\Delta x = \frac{b-a}{n}$.
The Absolute Difference Rule
What if the curves cross? If we simply integrate $f-g$ while $g$ is actually above $f$, we will get a negative result. To ensure we always calculate the magnitude of the area, we use the absolute value:
$$A = \int_a^b |f(x) - g(x)| dx$$
🎯 The Area Formula Theorem
If $f$ and $g$ are continuous functions and $f(x) \ge g(x)$ for all $x$ in $[a, b]$, the area $A$ of the region bounded by $y = f(x)$, $y = g(x)$, $x = a$, and $x = b$ is:
$$A = \int_a^b [f(x) - g(x)] dx$$
Example 1: Exponential vs. Linear
Find the area bounded above by $y = e^x$, below by $y = x$, from $x = 0$ to $x = 1$.
$$A = \int_0^1 (e^x - x) dx = [e^x - \frac{1}{2}x^2]_0^1 = (e - \frac{1}{2}) - (e^0 - 0) = e - 1.5 \approx 1.218$$