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

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7 5 area between two curves

7.5Area Between Two Curves

Find Area Between 2 Curves

Find Consumer Surplus

Find Producer Surplus


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


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)


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.


DEFINITION:

The equilibrium point, (xE, pE),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.


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,



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,



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.


Example (continued):

a) To find the equilibrium point, set D(x) = S(x) and solve.

Thus, xE = 2. To find pE, substitute xE into either D(x) or S(x) and solve.


Example (continued):

If we choose D(x), we have

Thus, the equilibrium point is (2, $9).


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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