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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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)
Second, find the points of intersection by setting f (x) = g(x) and solving.
2
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1
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(
2
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x
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x
d
x
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3
x
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x
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3
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2
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3
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Example (concluded):
Lastly, compute the integral. Note that on [0, 2], f (x) is the upper graph.
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.
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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s
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(
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(
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1
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Example: Find the consumer surplus for the demand function given by
When x = 3, we have Then,
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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2
3
1
5
(
3
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x
d
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=
+
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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.
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.
b) The consumer surplus at the equilibrium point is
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2
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=
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+
+
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1
8
3
2
3
0

+
+

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3
3
3
»
$
7
.
3
3
Example (concluded):
b) The producer surplus at the equilibrium point is