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Integration. area undera curve


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Integration. area undera curve

  1. 1. Rules of Integration ByOladokun Sulaiman Olanrewaju
  2. 2. 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
  3. 3. 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.
  4. 4. 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
  5. 5. 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
  6. 6. 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
  7. 7. 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).
  8. 8. 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
  9. 9. 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
  10. 10. 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