Calculus / Unit 9: Applications of Integrals
9.1 Area Between Two Curves
Notes
If $f(x) \geq g(x)$ on $[a, b]$, the area between the curves is $\displaystyle \int_a^b \left(f(x) - g(x)\right)dx$ (top minus bottom).
- Example: between $y = x$ and $y = x^2$ on $[0, 1]$, the line is on top. Area $\displaystyle = \int_0^1 (x - x^2)\,dx = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$.
- Always subtract the lower curve from the upper curve so the result is positive.
Practice
Find the area between $y = x$ and $y = x^2$ from $x = 0$ to $x = 1$.
$\displaystyle \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$.
Find the area between $y = 4$ and $y = x^2$ from $x = -2$ to $x = 2$.
$\displaystyle \int_{-2}^{2} (4 - x^2)\,dx = \left[4x - \frac{x^3}{3}\right]_{-2}^{2} = 2\left(8 - \frac{8}{3}\right) = \frac{32}{3}$.
On $[a, b]$, $f(x) \geq g(x)$. Which integral gives the area between them?
Subtract the bottom function from the top function.
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