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Unformatted text preview: 6 Integration (A.4) 6.1 Indefinite Integral • Consider a continuous function f ( x ) , where f ( x ) > 0 for all x . • Consider the area under the graph of y = f ( x ) from a certain point a to another point x and denote it by A ( x ; a ) . • What is the derivative of A ( x ; a ) at x > a ? • It is lim h → A ( x + h ; a )- A ( x ; a ) h • A ( x + h ; a )- A ( x ; a ) ≈ f ( x ) · h • Hence, lim h → A ( x + h ; a )- A ( x ; a ) h = f ( x ) . Figure 12: Area under a graph • A function F ( x ) whosederivative is a given function f ( x ) is called an antideriva- tive or the indefinite integral of f ( x ) , and written as F ( x ) = integraldisplay f ( x ) dx . • In the previous example, we have A ( x ; a ) = F ( x )+ C for some constant C . • Here are some examples. • integraltext x n dx = 1 n + 1 x n + 1 + C • integraltext ( f + g ) dx = integraltext fdx + integraltext gdx • integraltext ( f prime · g + f · g prime ) dx = f · g + C 18 Figure 13: Definite integral of a function 6.2 Definite Integral • Consider the area A under the graph of y = f ( x ) from a certain point b to another point c > b . • Pick a point a such that a < b < c . • Then A = A ( c ; a )- A ( b ; a ) or A = F ( c )- F ( b ) . • We define the definite integral of f ( x ) from b to c to be F ( c )- F ( b ) and write as F ( c )- F ( b ) = integraldisplay c b f ( x ) dx Economic Example: Integration of MC • Consider a firm with the following cost curves and a horizontal demand curve. • The profit is maximized at the output level where the marginal revenue ( = p ) equals the marginal cost. • The shaded area S can be written as S = pq *- integraltext q * MC ( q ) dq = pq *- TVC ( q * ) , because integraltext q * MC ( q ) dq = C ( q * )- C ( ) = TVC ( q * ) ....
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- Spring '11