Definitions [1]
Definition: Symmetrical Area
If a curve is symmetrical about a line, axis, or origin, then the total area can be obtained by finding the area of one symmetrical part and multiplying it by the number of such parts.
Key Points
Key Points: Area Under Simple Curves
| Case | Standard Form | Area Formula |
| Region above x-axis | y = f(x) | \[A = \int_{a}^{b} y \, dx\] |
| Region bounded by y-axis | x = g(y) | \[A = \int_{c}^{d} x \, dy\] |
| Curve below x-axis | y = f(x) < 0 | \[A = \left\vert \int_{a}^{b} f(x) \, dx \right\vert\] |
| Curve crossing x-axis | Mixed signs | \[A = \vert A_1 \vert + A_2\] |
Key Points: Area Bounded by Two Curves
- A = ∫ (upper − lower) dx
- Find intersection points → solve f(x) = g(x)
- If the graph crosses → split into parts
- Final area = sum of all parts
