Integration. area undera curve
Upcoming SlideShare
Loading in...5
×
 

Integration. area undera curve

on

  • 811 views

 

Statistics

Views

Total Views
811
Views on SlideShare
811
Embed Views
0

Actions

Likes
0
Downloads
29
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

Integration. area undera curve Integration. area undera curve Presentation Transcript

  • Rules of Integration ByOladokun Sulaiman Olanrewaju
  • Area Under A Curve• The area under a curve is the area bounded by the curve y = f(x), the x-axis and the vertical lines x = a and x = b. y y = f(x) Suppose we want to know the area of the shaded region x x=a x=b
  • Area and the Definite Integral• If a function has only positive values in an interval [a,b] then the area between the curve y=f(x) and the x-axis over the interval [a,b] is expressed by the definite integral, b a f ( x)dx• It is called the definite integral because the solution is an explicit numerical value.
  • f(x) < 0 for some interval in [a,b] b  a f ( x)dx  Area of R1  Area of R2  Area of R3 y y = f(x) R3 R1 x R2 x=a x=b
  • y To find the area y = f(x) below the x- axis R3 R1 place a negative sign x R2 before the x=b x=a x=c integral x=dAlternatively,Area of R1  Area of R2  Area of R3 c d b  f ( x)dx   f ( x)dx   f ( x)dx a c d
  • Area Between Two CurvesLet f and g be continuous functions such that f(x)>g(x)on the interval [a,b]. Then the area of the regionbounded above by y=f(x) and below by y=g(x) on [a,b]is given by y y = g(x)  f ( x)  g ( x)dx ba y = f(x) x x=a x=b
  • Fundamental Theorem of Calculus• If f is continuous on [a,b], then the definite integral is b a f ( x)dx  F (b)  F (a)where F(x) is any antiderivative of f on [a,b]such that F ( x)  f ( x).
  • Evaluating Definite Integrals bTo find  f ( x)dx aFirst find the indefinite Integral  f ( x)dx F ( x)  CThen find F (a) and F (b). b bFinally,  f ( x)dx  F ( x)  F (b)  F (a) a a
  • Consumer SurplusPrice, p Consumer surplus is a measure of consumer welfare. The area of this shaded region is x* Consumer 0 D( x)dx  p * x * Surplus p = p* Demand function p = D(x) 0 x* Quantity, x
  • Producer SurplusPrice, p Supply function p = S(x) Producer surplus is the area of this shaded region which is x* p * x *  S ( x)dx 0 p = p* Producer Surplus Demand function p = D(x) 0 x* Quantity, x