Integration Course Manual Indefinite Integration 7.-7. Definite Integration 7.-7.4 Jacques ( rd Edition) Indefinite Integration 6. Definite Integration 6. y F (x) x n + c dy/dx F`(x) f(x) n x n- Given the derivative f(x), what is F(x)? (Integral, Anti-derivative or the Primitive function).
Just as f(x) derivative of F(x) F ( x) f ( x) dx Example F ( x) x dx x + c cconstant of integration (since derivative of c) of course, c may be.., but it may not check: if yx + c then dy/dx x or if c, so yx then dy/dx x How did we integrate f(x)?
Rule of Integration: F( x) x n dx n + x n+ + c Examples F ( x) x dx x + check: if y / x + c then dy/dx x c F ( x ) dx.dx x dx x + check: if y x + c then dy/dx c
Rule of Integration: F( x) af ( x) dx a f ( x) dx Examples F ( x ) x dx x dx..x + c x + c F ( x ) a.dx a dx ax + c F ( x ) 4dx 4 dx 4x + c
Rule of Integration: F( x) [ f ( x) + g( x) ] dx f ( x) dx + g( x) dx Example [ ] x + x dx x dx+ x dx x + x c F ( x) +
Calculating Marginal Functions MR d( TR) dq MC d ( TC) dq Given MR and MC use integration to find TR and TC ( Q) MR( Q) TR. dq ( Q) MC( Q) TC. dq
Marginal Cost Function Given the Marginal Cost Function, derive an expression for Total Cost? MC f (Q) a + bq + cq TC( Q ) ( ) a + bq cq + dq TC( Q ) a c Q dq dq + bq dq + b c TC ( Q ) aq + Q + Q + F the constant of integration If Q, then TCF F Fixed Cost.. F
Another Example MC f (Q) Q + 5 If Total Cost when production is, find TC function? ( Q ) TC( Q ) + 5 dq TC( Q ) Q dq + 5 dq TC ( Q ) Q + 5Q + F the constant of integration If Q, then TC F Fixed Cost So if TC then, F TC( Q ) Q + 5Q +
Another Example Given Marginal Revenue, find the Total Revenue function MR f (Q) Q ( Q) TR( Q ) dq TR( Q ) dq QdQ TR ( Q ) Q Q + c the constant of integration c
Example: Given MCQ 6Q + 6; MR Q; and Fixed Cost. Find total profit for profit maximising firm when MRMC? Solution: ) Find profit max output Q where MR MC MRMC so Q Q 6Q + 6 gives Q Q 8 (Q - 4)(Q + 8) so Q +4 or Q - Q +4 ) Find TR and TC TR( Q ) Q dq ( ) TR( Q ) dq QdQ TR ( Q ) Q Q + so TR Q Q c
MC f (Q) Q 6Q + 6 ( ) Q 6Q TC( Q ) + 6 dq TC( Q ) Q dq 6QdQ + 6 dq TC ( Q ) Q Q + 6Q + F F Fixed Cost (from question) so. TC( Q ) Q Q + 6Q. Find profit TR-TC, by substituting in value of q* when MR MC Profit TR TC TR if q*4: (4) - 4 88-6 7 TC if q* 4: / (4) (4) + 6(4) / (64) 48 + 4 8 / Total profit when producing at MRMC so q*4 is TR TC 7-8 / 5 /
NOTE: Given a MR and MC curves - can find profit maximising output q* where MR MC - can find TR and TC by integrating MR and MC - substitute in value q* into TR and TC to find a value for TR and TC. then.. - since profit TR TC Can find (i) profit if given value for F or (ii) F if given value for profit
Definite Integration The definite integral of f(x) between values a and b is: [ F x) ] b b ( a f ( x) dx F( b) F( a) Example x dx a x () () 7 6 dx [ ] 6 x (6) ()
b The definite integral f ( x )dx a can be interpreted as the area bounded by the graph of f(x), the x-axis, and vertical lines xa and xb f(x) a b x
The Consumer Surplus Difference between value to consumers and to the market. P x Demand Curve: P f(q) P a Consumer Surplus Q Q CS(Q) oq ax - oq ap CS( Q) Q D( Q) dq PQ
Producer Surplus Difference between market value and total cost to producers P P y a Supply Curve: P g(q) Producer Surplus Q Q PS(Q) oq ap - oq ay PS( Q ) PQ Q S(Q )dq
examples.. Find a measure of consumer surplus at Q 5, for the demand function p 4Q Solution If Q 5, then p 4(5) CS( Q) Q D( Q) dq PQ P P Demand Curve: P f(q) 4Q Consumer Surplus Q 5 7.5 Q
Entire area under demand curve between and Q 5: 5 ( 4Q )dq [ ] Q Q ( 5 ) ( 5 ) total revenue area under price line (p ), between Q and Q 5 is p Q So CS p Q (*5) 5 5
Example : If p + Q is the supply curve, find a measure of producer surplus at Q 4 Solution If Q 4, then p + 6 9 PS( Q ) PQ Q S(Q )dq P Supply Curve: P g(q) + Q P 9 Producer Surplus Q 4 Q
Entire area under supply curve between Q and Q 4.. 4 4 4 4 + + + ) ( ) ( Q Q )dq Q ( total revenue area under price line (p 9), between Q and Q 4 is p Q 76 So PS p Q / 76 / 4 /
Manual, Topic 7 Q. A profit maximising firm has MR 4 Q and MC Q Q + 6. How much will it produce? What level of fixed costs would make the firm make zero profits? Step : set MRMC and find output that maximises profit, q* Q Q + 6 4 Q Q 7Q 8 Solve the quadratic for value of Q using b ± b 4( a)( c) Q formula ( a) : a, b-7, c-8 7 ± 49 4()( 8) 7 ± 9 Q () so Q (inadmissible) or Q 8 Thus 8 units produced by profit max firm
Step : integrate MR and MC to find TR & TC, and thus profits π TR TC TR MR. dq + ( 4 Q ) dq 4Q Q c In this case, the constant of integration c, since the firm makes no revenue when Q TC MC. dq 6 + ( ) Q Q + 6 dq Q 5Q + Q F F, the constant of integration Fixed Costs π 4Q Q Q + 5Q 6Q 7 π Q + Q + 8Q F F
Step : substitute in q* to TR and TC to get profit max values when producing q* Substituting in Q 8 for profit max. π 7 () 8 + () 8 + 8() 8 7 + 4 + 64 7 F Step 4: Set profit (thus TR TC ), & solve for F F 7 Setting π, gives 7 F Thus, value of F at π is
Q4 (b): A firm which has no fixed costs has MC and MR given as follows: MCQ 6Q + 6; MR Q; Find total profit for profit maximising firm when MRMC? Solution: ) Find profit max output Q where MR MC Q Q 6Q + 6 gives Q Q 8 Solve quadratic for Q, by using formula, or (Q - 4)(Q + 8) so Q +4 or Q - so Q +4 (since Q- inadmissable) ) Find TR and TC TR( Q ) ( Q) dq TR( Q ) dq QdQ TR ( Q ) Q Q + c
TR c when Q; but TR when Q ; so therefore c so TR Q Q MC f (Q) Q 6Q + 6 ( ) Q 6Q TC( Q ) + 6 dq TC( Q ) Q dq 6QdQ + 6 dq TC ( Q ) Q Q + 6Q + F F Fixed Cost (from question) so. TC( Q ) Q Q + 6Q
. Find profit TR-TC, by substituting in value of q* when MR MC Profit TR TC TR if q*4: (4) - 4 88-6 7 TC if q* 4: / (4) (4) + 6(4) / (64) 48 + 4 8 / so total profit when producing at MRMC at q*4 is TR TC 7-8 / 5 /
Q5. The demand and supply functions for a good are given by the equations P Q + 4 and P Q + respectively. Determine the equilibrium price and quantity and calculate the consumer and producer surplus at equilibrium. At equilibrium Q + 4 Q + Q So equilibrium Q * 4 Thus equilibrium P * 4 + 6 P 4 CS S PS P * 6 D Q * 4 7 Q Consumer Surplus
Difference between value to consumers and to the market. Area above price line and under Demand curve CS CS Q* 4 ( ) P * Q * D Q dq ( Q + 4) ( 6)( 4) dq CS CS [ Q + ] 4 Q 4 4 [( ( ) ( )) ( ( ) ( ))] 4 + 4 4 + 4 4 CS 6 + 56 4 6
Producer Surplus Difference between market value and total cost to producers area below price line and above Supply curve PS PS PQ Q S ( Q) 4 ()() 4 ( Q + ) 6 dq. dq PS 4 Q + Q 4 PS 4 PS 4 8 8 8 ( 4) + ( 4) () + () Total Surplus CS + PS 6 + 8 4