1 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Chapter 8: Integration and Applications u Definite Integration and Area: Figures 8.6, 8.7, 8.8 Slides 2, 3, 4, 5 u Examples on definite integration: Slide 6 u Consumer Surplus: Slide 7 u Worked Example 8.12(b) Figure 8.12 Determine Consumer surplus: Slide 8, 9 u Producer Surplus. Figure 8.13: Slide10,11 u Figure 8.15 Producer surplus for Worked Example 8.13 (b): Slide 12

2 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Integration and area u Figure 8.6 Area under the curve approximately equal to the sum of areas of rectangles. Area yxyxyxyx    n123  Area i a i b i y x      x  x  x  x y 1 y 2 y 3 y n-1 x=a x=b y =f (x) y x

3 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Integration and area  Figure 8.7 Decreasing the size of x gives a better approximation to area   x = a x = b widths,  x  0 y= f(x) y x

4 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Use Integration to determine the exact area u When dealing with infinitely small changes, the notations change: u As in differentiation, replace x by dx to indicate that x is approaching zero. u Also replace the summation sign,, by the integral sign u Therefore, as x tends to 0 u is called the definite integral of y w.r.t x between the limits of integration, a and b. 

5 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Integration and area u Figure 8.8 Area under the curve is determined exactly by integration. Area = f ( )d xx x a x b    y =f(x) x=a x=b y x Net area = fdFFF()()()()xxxba xa xb xa xb       (8.9)

6 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd u Examples of definite integration

7 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Consumer Surplus u Consumer Surplus at (8.12) PP  0 (the corresponding quantity is Q ) is area under the demand function (between Q = 0 and Q 0 ) - area of rectangle (P 0 Q 0 ) 0 P P 0 Q Q 0 Consumer surplus Demand function 0 RectangleP 0  Q 0 Area under function Consumer surplus

8 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Consumer Surplus u Consumer surplus for P = Q Area under demand function = 750 Area of rectangle = 500 CS = = 250 P 100 P 0 =50 Q 20 Q 0 =10 Consumer surplus Actual expenditure P = Q 0 Alternatively, CS = Area of shaded triangle = 0.5(10)(50) = 250

9 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Consumer Surplus for a non-linear demand function u Figure 8.12 Consumer surplus for Worked Example 8.12(b) Area under demand function = Area of rectangle = 60 CS = = P 50 P = 20 Q = 3 Q Consumer surplus P Q   The area under the demand function must be calculated by integrating the demand function fromQ = 0 to 3

10 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Producer Surplus Producer surplus at PP  0 area of rectangle - area under the supply function PQQ Q QQ 00   (supply functionon)d =0 = 0 = : ( Q = Q 0 ) P Q Q 0 Producer surplus Area under the supply function P 0 Supply function 0

11 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Producer Surplus Producer surplus Area under supply function = Area of rectangle = 385 PS = = See introduction to PS for calculations based on integration P 20 Q Q 0 =7 Producer surplus Revenue the producer is willing to accept P 0 = 55 P = Q 0

12 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Producer Surplus for a non-linear supply function Figure 8.15 Producer surplus for Worked Example 8.12 (b) Area under supply function = Area of rectangle = 160 PS = = P P 0 = 40 0 Q Q 0 =4 Producer surplus Revenue the producer is willing to accept P =Q 2 + 6Q The area under the supply function must be calculated by integrating the supply function fromQ = 0 to 4