Area Between Curves

Definition

Area Between Curves is the application of the definite integral to calculate the size of a region bounded by two or more functions. It is found by integrating the difference between the “upper” and “lower” boundaries of the region.

Why It Matters

It allows us to calculate the net difference between two competing processes, such as total profit or net energy gain. Without it, we could see individual trends but never the total magnitude of the gap between them.

Core Concepts

• Vertical Rectangles ($dx$): $A = \int_a^b [f(x) - g(x)] dx$, where $f(x)$ is the top curve.

  • How to read: “The area A equals the integral from a to b of the quantity f of x minus g of x, with respect to x.”
  • Meaning: Stack thin vertical strips; height is top minus bottom. Ensure $f(x) \geq g(x)$ on $[a,b]$ or split where they cross.

• Horizontal Rectangles ($dy$): $A = \int_c^d [f(y) - g(y)] dy$, where $f(y)$ is the rightmost curve.

  • How to read: “The area A equals the integral from c to d of the quantity f of y minus g of y, with respect to y.”
  • Meaning: Use when curves are easier as $x$ as a function of $y$—right minus left.

• Intersection Points: The limits of integration ($a, b$) are often found by solving $f(x) = g(x)$.

  • How to read: “f of x equals g of x.”
  • Meaning: Intersections bound the region and may require splitting the integral if curves swap top/bottom.

Connected Concepts

• Area of a Triangle
• Conceptual Foundation of Space Curves
• Level Curves
• Level Surfaces