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 ba 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