Lecture 1 Intergration

Introduction Help answer the question, What is f if we have been told that its derivative is equal to 2x , so that f’(x)=2x ? Since the derivative of x2 is 2x, so does x2 + c where c is an arbitrary constant, since additive constants disappear with differentiation. This shows that if: f’(x)=2x ↔ f(x)=x2 + c Suppose f(x) and F(x) are two functions of x having the property that f(x)=F’(x) Since we pass from F to f by taking the derivative, the reverse process of passing from f to F could appropriately be called taking the Antiderivative. F is called an indefinite integral of f

Mathematical Notation As notation for an indefinite integral of f we use ∫f(x) dx Two functions having the same derivative must differ throughout an interval must differ by a constant, so we write ∫f(x)dx=F(x)+C when F’(x)=f(x)……….(1) The symbol ∫ is the integral sign and the function f(x) appearing in (1) is the integrand, we write dx to indicate that x is the variable of integration.

Mathematical Notation Cont. We read (1) this way : the indefinate intergral of f(x) w.r.t x is F(x) plus a constant. We call it an indefinite integral because F(x)+C is not to be regarded as one definite function but as a whole class of functions having the same derivative f. It follows that the derivative of the indefinite integral is equal to the integrand: d/dx∫f(x)dx=f(x)………….(2) Also ∫F’(x)dx=F(x)+C………….(3) Thus integration and differentiation cancel each other out.

Some Important Integrals Because the derivative of xa+1 /a+1 is xa it follows that ∫xa dx=1/a+1 xa + C where a≠1………(4) Find: ∫xdx, ∫1/x3 dx, ∫√x dx, ∫1/x dx, ∫eax dx

Lecture 2: Some General Rules ∫af(x)dx=a∫f(x)dx (a is a constant)……..(5) ∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx……………..(6) Examples: Evaluate ∫3x4 +5x2 +2dx and ∫3/x – 8e-4x dx (x>0) Class Exercise: Evaluate ∫B/r2.5 dr, ∫(-16)dx, ∫(x3 +x ) dx

Area under a curve and Definite Integration Use integration to compute the area under a graph of a continuous and non-negative function f over the interval [a,b]. Illustration to be provided in class To find the area of the curve f over the interval [a,b], simply integrate f w.r.t x over the given interval i.e ∫b af(x) dx

The Definite Integral Let f be a continuous function defined in the interval [a,b] and has a derivative with F’(x)=f(x) for every x in [a,b] Then the difference F(b)-F(a) is called the definite integral of f over [a,b] which is therefore a number that depends only on the function f and the numbers a and b denoted by: ∫ab f(x)dx the numbers a and b are called respectively, the lower and upper limit of integration. The difference F(b)-F(a) is denoted by │a bF(x) or by [F(x)]ab

The definite Integral Cont.. Thus the definition of the Definite Intergral is: ∫ab f(x)dx= │a bF(x) =F(b)-F(a)……….(7) Where F is any function satisfying F’(x)=f(x) for all x in [a,b] Class Exercise: Evaluate ∫52 e2x dx and ∫2-2 (x –x3 –x5 )dx Compute the areas under the graphs of f(x)=x10 Over the [0,1]

Properties of Definite intergrals If f is a continuous function in an interval that contains a, b and c, then: ∫ab f(x)dx=- ∫ba f(x)dx ∫aa f(x)dx=0 ∫ab α f(x)dx= α ∫ab f(x)dx (α is an arbitrary constant) ∫ab f(x)dx= ∫ac f(x)dx+ ∫cb f(x)dx

Lecture 3 Economic Application

From Marginal to a Total Function Given a total function (e.g. a total cost function), the process of differentiation can yield the marginal function (e.g. marginal cost function) Being the opposite of integration, integration enables us, conversely, to infer total function from a given marginal function Example 1: if the Marginal cost (MC) of a firm is the following function of output, C’(Q)=2e0.2Q , and if the fixed cost is CF =90. Find the total-cost function C(Q)

From Marginal to a Total Function Cont… Solution: Integrate C’(Q) w.r.t Q ∫ 2e0.2Q dQ=2(1/0.2)e0.2Q +C =10e0.2Q +C …………..(*) But when Q=0, C(0)=CF =90 Setting Q=0 in * yields C(0)= 10e0.2(0) +C =10+C 90-10=C 80=C Therefore: C(Q)= 10e0.2Q + 10

From Marginal to a Total Function Cont… Example 2: if the marginal propensity to save (MPS) is the following function of income, S’(Y)=0.3-0.1Y-1/2 , and if the aggregate saving S is nil when Y is 81, find the savings function S(Y). Solution: To be provided in class Class Exercise: suppose that the investment flow is described by the equation I(t)=3t1/2 and that the initial capital stock at time t=0; is K(0). What is the time path of capital K(t)? Hint: K(t) is the integral of I(t)