3.1 Integration as the Inverse of Differentiation
Integration is the reverse process of differentiation. If $\frac{d}{dx}[F(x)] = f(x)$, then the indefinite integral of $f(x)$ with respect to $x$ is:
$$\int f(x) \, dx = F(x) + c$$
where $c$ is the constant of integration.
3.2 Indefinite Integral
Basic Integration Rules
For algebraic functions where $a$, $k$, and $n$ are real numbers, and $n \neq -1$:
- Constant Rule: $\int a \, dx = ax + c$
- Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$
- Scalar Multiple Rule: $\int k f(x) \, dx = k \int f(x) \, dx$
- Sum and Difference Rule: $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$
Integration of Composite Functions $(ax + b)^n$
For linear composite algebraic functions where $n \neq -1$ and $a \neq 0$:
$$\int (ax + b)^n \, dx = \frac{(ax + b)^{n+1}}{a(n + 1)} + c$$
Finding the Constant of Integration $c$
The constant $c$ can be determined when a specific point $(x, y)$ on the curve is given. Integrating the gradient function $\frac{dy}{dx}$ yields the equation of the curve $y = F(x) + c$.
3.3 Definite Integral
Fundamental Theorem of Calculus
The definite integral of a continuous function $f(x)$ over the closed interval $[a, b]$ is given by:
$$\int_{a}^{b} f(x) \, dx = [F(x)]_{a}^{b} = F(b) - F(a)$$
Properties of Definite Integrals
- $\int_{a}^{a} f(x) \, dx = 0$
- $\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$
- $\int_{a}^{b} k f(x) \, dx = k \int_{a}^{b} f(x) \, dx$
- $\int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx$
3.4 Applications of Integration
Area under a Curve
- Area bounded by curve $y = f(x)$, x-axis, and lines $x = a$, $x = b$:
$$A = \int_{a}^{b} y \, dx \quad \text{or} \quad A = \left| \int_{a}^{b} f(x) \, dx \right| \quad \text{(if below x-axis)}$$
- Area bounded by curve $x = g(y)$, y-axis, and lines $y = c$, $y = d$:
$$A = \int_{c}^{d} x \, dy$$
- Area between two curves/line and curve:
$$A = \int_{a}^{b} [f(x) - g(x)] \, dx$$
Volume of Revolution
- Volume generated when region bounded by $y = f(x)$ is rotated $360^\circ$ about the x-axis:
$$V = \pi \int_{a}^{b} y^2 \, dx$$
- Volume generated when region bounded by $x = g(y)$ is rotated $360^\circ$ about the y-axis:
$$V = \pi \int_{c}^{d} x^2 \, dy$$