📘 Area Between Two Curves

🔹 Basic Idea

Area between curves = top function − bottom function

$$\text{Area} = \int_a^b \big[ f(x) - g(x) \big]\,dx$$

• $f(x)$ = top curve
• $g(x)$ = bottom curve

---

🔹 Steps

1. Find intersection points → limits $a$, $b$
2. Identify top and bottom functions
3. Compute:

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

---

🔹 Example

Find area between $f(x) = x^2$ and $g(x) = x$

Step 1: Intersection

$$x^2 = x \Rightarrow x(x-1)=0 \Rightarrow x=0,1$$

Step 2: Top - Bottom

• Top: $x$
• Bottom: $x^2$

Step 3: Compute

$$\int_0^1 (x - x^2)\,dx$$

Antiderivative:

$$= \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$$

---

🔹 Vertical Slices (in terms of $y$)

If using $y$:

$$\text{Area} = \int_c^d \big[ x_{\text{right}} - x_{\text{left}} \big]\,dy$$

---

🔹 Key Takeaways

• Always do top − bottom (or right − left)
• Find intersection points carefully
• Sketch if unsure