Unit 5: Integration of functions

1. Integration as Anti-differentiation

Definition: Integration is the reverse process of differentiation. If differentiating function F(x) gives you f(x), then integrating f(x) gives you back F(x) (plus a constant).

If d/dx [F(x)] = f(x), then ∫f(x)dx = F(x) + C

F(x) is called the antiderivative or indefinite integral of f(x).

C is the constant of integration. It is required because the derivative of any constant is zero. We lose this constant during differentiation, so we must add it back when integrating.

Example:
We know d/dx (x²) = 2x.
Therefore, ∫2x dx = x² + C.
(The original function could have been x², or x²+5, or x²-100. All have a derivative of 2x).

2. Basic rules of integration

These rules are the reverse of the differentiation rules.

3. Definite and indefinite integral

A. Indefinite Integral

This is what we have seen so far. It is the general antiderivative, which results in a function + C.

∫f(x)dx = F(x) + C

B. Definite Integral

A definite integral is an integral with limits of integration (a "from" value a and a "to" value b). It results in a single numerical value.

Geometrically, the definite integral represents the area under the curve f(x) from x = a to x = b.

The Fundamental Theorem of Calculus:

To evaluate the definite integral of f(x) from a to b:

∫_a^b f(x)dx = [ F(x) ]_a^b = F(b) - F(a)

1. Find the antiderivative F(x). (You can ignore the +C, as it will cancel out).
2. Plug in the upper limit b: F(b).
3. Plug in the lower limit a: F(a).
4. Subtract the two: F(b) - F(a).

Example: Find the area under the curve f(x) = 2x from x=1 to x=3.
∫₁³ 2x dx
1. Antiderivative F(x) = x²
2. F(b) = F(3) = (3)² = 9
3. F(a) = F(1) = (1)² = 1
4. F(b) - F(a) = 9 - 1 = 8.

4. Application of integration in economics

Integration allows us to go from a Marginal concept back to a Total concept.

Finding Total Cost from Marginal Cost

If you are given the Marginal Cost function MC(Q), you can find the Total Cost function TC(Q).

TC(Q) = ∫MC(Q) dQ = VC(Q) + FC

• When you integrate MC, you get the Total Variable Cost (VC), plus the constant of integration C.
• In economics, this constant C is not just any constant; it is the Fixed Cost (FC)—the cost incurred even when Q = 0.

Example:
Given MC(Q) = 2Q + 10, and Fixed Costs (FC) are 50. Find the Total Cost (TC) function.
1. Integrate MC:
TC(Q) = ∫(2Q + 10) dQ = 2(Q²/2) + 10Q + C = Q² + 10Q + C
2. Solve for C:
We know that TC(0) = FC.
TC(0) = (0)² + 10(0) + C = 50 => C = 50
3. Final Function:
TC(Q) = Q² + 10Q + 50

5. Consumer's surplus

Definition: Consumer's Surplus (CS) is the total benefit or "gain" consumers receive by paying the market price (P_e) for a good, when many of them were willing to pay a higher price.

It is the area below the demand curve and above the equilibrium price line, from Q=0 to Q=Q_e.

Calculating Consumer's Surplus:

CS = (Total area under Demand curve) - (Total amount spent by consumers)
Total amount spent = P_e * Q_e (a rectangle)
Total area under Demand = ∫₀^Qe D(Q) dQ

CS = ∫₀^Qe [D(Q) - P_e] dQ

... or, more simply:

CS = [ ∫₀^Qe D(Q) dQ ] - (P_e * Q_e)

6. Producer's surplus

Definition: Producer's Surplus (PS) is the total benefit or "gain" producers receive by selling at the market price (P_e), when many of them were willing to sell for a lower price.

It is the area above the supply curve and below the equilibrium price line, from Q=0 to Q=Q_e.

Calculating Producer's Surplus:

PS = (Total amount received by producers) - (Total area under Supply curve)
Total amount received = P_e * Q_e (a rectangle)
Total area under Supply = ∫₀^Qe S(Q) dQ

PS = ∫₀^Qe [P_e - S(Q)] dQ

... or, more simply:

PS = (P_e * Q_e) - [ ∫₀^Qe S(Q) dQ ]