1. Fundamental Theorem of Integral Calculus
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Geometrical Meaning
Geometrically, the definite integral $\int_{a}^{b} f(x) dx$ gives the area $A$ under the curve $y=f(x)$ (where $f(x)>0$), bounded by the X-axis and the vertical lines $x=a$ and $x=b$.
The exact area is evaluated using the formula:
$$ \int_{a}^{b} f(x) dx = \phi(b) - \phi(a) $$
where $\int f(x) dx = \phi(x)$.
2. Area Under the Curve
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2.1 Area Bounded by the X-axis
If the curve $y=f(x)$ is continuous in $[a, b]$ and $f(x) \ge 0$ in $[a, b]$, the area bounded by the curve $y=f(x)$, X-axis, and the lines $x=a$ and $x=b$ is given by:
$$ A = \int_{a}^{b} f(x) dx = \int_{x=a}^{x=b} y dx $$
Area bounded by the curve and the X-axis -
2.2 Area Bounded by the Y-axis
The area $A$, bounded by the curve $x=g(y)$, the Y-axis, and the horizontal lines $y=c$ and $y=d$ is given by:
$$ A = \int_{c}^{d} x dy = \int_{y=c}^{y=d} g(y) dy $$
Area bounded by the curve and the Y-axis
3. Area Between Two Curves
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Finding Area Between Two Intersecting Curves
Let $y=f(x)$ and $y=g(x)$ be the equations of two curves.
The required area $A$ bounded between them from $x=a$ to $x=b$ is given by $A = |A_1 - A_2|$, where:
- $A_1$ is the area under $y=f(x)$
- $A_2$ is the area under $y=g(x)$
$$ A = \left| \int_{a}^{b} f(x) dx - \int_{a}^{b} g(x) dx \right| $$
Area bounded between two continuous curves
4. Area with Negative Function Values
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Handling Regions Below the X-axis
If $f(x) \le 0$ or $f(x) \ge 0$ in $[a, b]$, the geometric area enclosed is given by the absolute value: $\left| \int_{a}^{b} f(x) dx \right|$.
If the curve crosses the X-axis at $x=t$ (dividing the area into $A_1$ below the axis and $A_2$ above the axis), the total geometric area must be evaluated in parts to avoid cancellation:
$$ A = A_1 + A_2 = \left| \int_{a}^{t} f(x) dx \right| + \left| \int_{t}^{b} f(x) dx \right| $$
Splitting the integral when the curve crosses the axis
5. Tracing of Curve (Symmetry)
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Symmetry Rules
X-axis Symmetry: The X-axis is an axis of symmetry for a curve $C$, if $(x, y) \in C \iff (x, -y) \in C$. (Occurs if all powers of $y$ in the equation are even).
Y-axis Symmetry: The Y-axis is an axis of symmetry for a curve $C$, if $(x, y) \in C \iff (-x, y) \in C$. (Occurs if all powers of $x$ in the equation are even).
Origin Symmetry (Both Axes): If replacing $x$ and $y$ with $-x$ and $-y$ respectively leaves the equation unchanged, the curve is symmetric about both the X-axis and Y-axis (or opposite quadrants).