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7.5 Area Between Two Curves Find Area Between 2 Curves Find Consumer Surplus Find Producer Surplus.

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Presentation on theme: "7.5 Area Between Two Curves Find Area Between 2 Curves Find Consumer Surplus Find Producer Surplus."— Presentation transcript:

1 7.5 Area Between Two Curves Find Area Between 2 Curves Find Consumer Surplus Find Producer Surplus

2 Area between 2 curves Let f and g be continuous functions and suppose that f (x) ≥ g (x) over the interval [a, b]. Then the area of the region between the two curves, from x = a to x = b, is

3 Example: Find the area of the region that is bounded by the graphs of First, look at the graph of these two functions. Determine where they intersect. (endpoints not given)

4 Example (continued): Second, find the points of intersection by setting f (x) = g (x) and solving.

5 Example (concluded): Lastly, compute the integral. Note that on [0, 2], f (x) is the upper graph. (2 x  1)  ( x 2  1)     0 2  dx  (2 x  x 2 ) 0 2  dx  x 2  x 3 3       0 2  2 2         0 2         4  8 3  0  0  4 3

6 DEFINITION: The equilibrium point, (x E, p E ), is the point at which the supply and demand curves intersect. It is that point at which sellers and buyers come together and purchases and sales actually occur.

7 DEFINITION: Suppose that p = D(x) describes the demand function for a commodity. Then, the consumer surplus is defined for the point (Q, P) as

8 Example: Find the consumer surplus for the demand function given by When x = 3, we have Then, Consumer Surplus  Dx  dx  x E  p E 0 x E   ( x  5) 2 dx  3    ( x 2  10 x  25) dx 0 3   12

9 Example(concluded):

10 DEFINITION: Suppose that p = S(x) is the supply function for a commodity. Then, the producer surplus is defined for the point (Q, P) as

11 Example : Find the producer surplus for Find y when x is 3. When x = 3, Then, Producer Surplus  x E  p E  Sx  dx 0 x E   3  15  ( x 2  x  3) dx 0 3 

12 Example (continued):

13 Example: Given find each of the following: a) The equilibrium point. b) The consumer surplus at the equilibrium point. c) The producer surplus at the equilibrium point.

14 Example (continued): a) To find the equilibrium point, set D(x) = S(x) and solve. Thus, x E = 2. To find p E, substitute x E into either D(x) or S(x) and solve.

15 Example (continued): If we choose D(x), we have Thus, the equilibrium point is (2, $9).

16 Example (continued): b) The consumer surplus at the equilibrium point is

17 Example (concluded): b) The producer surplus at the equilibrium point is


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