Area Between Curves

$A = \int_a^b [f(x) - g(x)]\,dx$

where $f(x) \geq g(x)$ on $[a,b]$ (top minus bottom).

Horizontal Slicing

$A = \int_c^d [f(y) - g(y)]\,dy$

(right minus left)

1. Sketch the curves.
2. Find intersection points (set equal).
3. Determine which is on top.
4. Integrate (top − bottom).

Area between $y = x^2$ and $y = x + 2$.

Intersection: $x^2 = x + 2$ → $x^2 - x - 2 = 0$ → $(x-2)(x+1) = 0$ → $x = -1, 2$.

$A = \int_{-1}^{2} [(x+2) - x^2]\,dx = [\frac{x^2}{2} + 2x - \frac{x^3}{3}]_{-1}^{2} = (2+4-\frac{8}{3}) - (\frac{1}{2}-2+\frac{1}{3}) = \frac{9}{2}$.