Ch 08 - Application of Integrals12th Mathematics

Component Anatomy:

• $y = f(x)$: The boundary curve.
• $[a, b]$: The interval boundaries on the x-axis.
• $y$: The height of the infinitesimal vertical strip.
• $dx$: The width of the vertical strip.

Boundary Conditions / Domain:

• The curve $f(x)$ must be continuous on the interval $[a, b]$.
• To represent purely positive geometric area without taking absolute values, $f(x) \ge 0$ must hold for all $x \in [a, b]$.

Component Anatomy:

• $x = g(y)$: The curve expressed as a function of $y$.
• $[c, d]$: The interval boundaries on the y-axis.
• $x$: The horizontal length of the infinitesimal strip.
• $dy$: The vertical thickness of the strip.

Boundary Conditions / Domain:

• The curve $g(y)$ must be continuous on $[c, d]$.
• To represent purely positive geometric area without absolute values, $g(y) \ge 0$ must hold for all $y \in [c, d]$.

• Concept: Definite integrals strictly compute algebraic area. If a curve lies strictly below the x-axis (where $f(x) < 0$), the integral evaluates to a negative number.
• The Rule: Because geometric physical area cannot be negative, you MUST take the absolute value (modulus) of the definite integral for that specific region.
• Formula: $A = \left| \int_{a}^{b} f(x) dx \right|$