BCA TU, Math II – Unit 4: Integration and Its Applications

Integration is the reverse process of differentiation. While differentiation deals with the rate of change, integration deals with accumulation — such as finding area under curves, total distance, volume, and surface area.

1. Riemann Integral

The Riemann integral defines integration as the limit of sums of areas of rectangles under a curve.

If a function $f(x)$ is continuous on $[a, b]$ , then:

$\int_a^b f(x) , dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x$

where $\Delta x = \frac{b-a}{n}$ .

2. Fundamental Theorem of Calculus (Without Proof)

Connects differentiation and integration:

If $F(x)$ is an antiderivative of $f(x)$ , then:

$\int_a^b f(x) , dx = F(b) - F(a)$

Differentiation and integration are inverse processes.

3. Techniques of Integration

Basic Rules

$\int k f(x) , dx = k \int f(x) , dx$

$\int (f(x)+g(x)) , dx = \int f(x) , dx + \int g(x) , dx$

Standard Integrals

$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$

$\int e^x dx = e^x + C$

$\int \frac{1}{x} dx = \ln|x| + C$

$\int \sin x , dx = -\cos x + C$

$\int \cos x , dx = \sin x + C$

Substitution Method

If $u = g(x)$ , then:

$\int f(g(x)) g'(x) dx = \int f(u) du$

Integration by Parts

$\int u v dx = u \int v dx - \int \left( \frac{du}{dx} \int v dx \right) dx$

Partial Fractions

Used for rational functions.

4. Definite and Improper Integrals

Definite Integral: $\int_a^b f(x) dx = F(b) - F(a)$
Improper Integrals: Extend to infinity or discontinuous functions.

Example:

$\int_1^\infty \frac{1}{x^2} dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^2} dx = 1$

5. Applications of Definite Integrals

Area under Curve: $A = \int_a^b f(x) dx$
Length of Arc (Rectification): $L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx$
Volume of Solid of Revolution:
Rotating curve $y=f(x)$ around x-axis: $V = \pi \int_a^b [f(x)]^2 dx$
Surface Area of Revolution: