Unit 3: Calculus (Part 2)

Integration & its Applications [Syllabus 3.6 – 3.9]

1. Indefinite Integrals

Integration is the reverse process of differentiation. If $$\frac{d}{dx}F(x) = f(x)$$, then $$\int f(x) dx = F(x) + C$$.

Power Rule $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$(where$$n \neq -1$$)
Logarithmic Rule $$\int \frac{1}{x} dx = \log|x| + C$$
Exponential Rule $$\int e^x dx = e^x + C$$
$$\int a^x dx = \frac{a^x}{\log a} + C$$
Properties 1. $$\int k \cdot f(x) dx = k \int f(x) dx$$
2. $$\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$$

2. Methods of Integration

By Substitution If the integral is of the form $$\int f(g(x))g'(x) dx$$:
Put $$g(x) = t \Rightarrow g'(x)dx = dt$$.
By Partial Fractions Used for rational functions $$\frac{P(x)}{Q(x)}$$.
Example: $$\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$
By Parts (Product Rule) For product of two functions $$u$$and$$v$$:
$$\int (u \cdot v) dx = u \int v dx – \int \left( \frac{du}{dx} \int v dx \right) dx$$
(Select ‘u’ using ILATE rule)

3. Definite Integrals & Area

Represents the area under the curve $$y=f(x)$$between limits$$a$$and$$b$$.

Fundamental Theorem If $$\int f(x) dx = F(x)$$, then:
$$\int_{a}^{b} f(x) dx = F(b) – F(a)$$
Area under Simple Curves Area bounded by curve $$y=f(x)$$, x-axis, $$x=a$$and$$x=b$$:
Area $$= \int_{a}^{b} y dx$$

4. Economic Applications

Finding Total Cost/Revenue From Marginal Cost (MC): $$C(x) = \int (MC) dx + k$$
From Marginal Revenue (MR): $$R(x) = \int (MR) dx$$
Consumer Surplus (CS) Gain to consumers who were willing to pay more than market price.
$$CS = \int_{0}^{x_0} f(x) dx – p_0 x_0$$
(Where $$f(x)$$is Demand curve,$$p_0$$is market price,$$x_0$$ is market quantity)
Producer Surplus (PS) Gain to producers who were willing to sell for less than market price.
$$PS = p_0 x_0 – \int_{0}^{x_0} g(x) dx$$
(Where $$g(x)$$ is Supply curve)
Equilibrium Point Point where Demand = Supply.
Solve $$f(x) = g(x)$$ to find equilibrium quantity ($$x_0$$) and price ($$p_0$$).
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