1. The "area under the curve" refers to the region between a graph of a function and the x-axis over a certain interval. 2. It is often calculated using definite integrals in calculus. 3. For a function $f(x)$, the area under the curve from $a$ to $b$ is given by the definite integral $$\int_a^b f(x)\,dx$$. 4. This integral sums up infinitely many infinitesimally small rectangles under the curve to find the total area. 5. If $f(x)$ is positive on $[a,b]$, the integral gives the exact area. 6. If $f(x)$ is negative, the integral gives a negative value representing area below the x-axis. 7. To find the total area regardless of sign, you can integrate the absolute value: $$\int_a^b |f(x)|\,dx$$. 8. This concept is fundamental in physics, economics, and probability for calculating quantities like distance, work, and probabilities.