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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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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

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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)

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

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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 2 3 3 0 2 0 3 3 4 8 3 0 0 4 3

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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.

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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

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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 4 0 3 ( x 2 10 x 25) dx 0 3 12

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Example(concluded):

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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

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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

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Example (continued):

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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.

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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.

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

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

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

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