Area Between Curves

Area Between Curves

• Definition: Area between two curves $f(x)$ and $g(x)$ from $x = a$ to $x = b$, where $f(x) \geq g(x)$: $$ A = \int_a^b [f(x) - g(x)] , dx $$
• For curves in terms of $y$ ($x = f(y)$ and $x = g(y)$), with $f(y) \geq g(y)$: $$ A = \int_c^d [f(y) - g(y)] , dy $$

Key Steps

• Find intersections of the curves to determine limits of integration.
• Determine which curve is on top (or to the right).
• Set up integral of top minus bottom (or right minus left).
• Integrate and evaluate.

Example (Enclosed Region)

• For curves $y = 9 - x^2$ and $y = x^2 + 1$ from $x = 0$ to $x = 3$:
  • Intersections: $9 - x^2 = x^2 + 1 \Rightarrow x = \pm 2$, so region from $x=0$ to $x=2$.
  • Top: $y = 9 - x^2$, bottom: $y = x^2 + 1$.
  • Area: $$ A = \int_0^2 [(9 - x^2) - (x^2 + 1)] , dx = \int_0^2 (8 - 2x^2) , dx $$

Handling Absolute Value

• For $y = 2 + |x-1|$, split at $x=1$ into two linear pieces.