Outline1Indefinite integrationRules for indefinite integration2Definite integration3Economic ApplicationsDerive TC from MCDerive TR from MRConsumer surplusProducer surplus4Other applications5Summary
Objectives•Use the rules of integration to find integrals•Find total functions from marginal functions•Find consumer surplus•Find producer surplus
LiteratureRenshaw, ch. 18
Integration•Indefinite integration - reverses differentiation, in the sameway the multiplication reverses division. Also known as theanti-derivative•Definite integration - allows us to calculate the area under afunction/curve (area between x-axis and curve)
Indefinite integral, p. 554-61•IfddxF(x) =f(x), thenRf(x)dx=F(x) +c•R= integral sign,f(x) is the integrand, c = constant ofintegration, dx indicates that we integrate w.r.t.x•Note theRf(x) just means integral sign f(x), whileRf(x)dxmeans that we have to integrate f(x) w.r.t. x. Notation isimportant here.•The relationship between indefinite integral and derivative:ddxZf(x)dx=f(x)The derivative of the indefinite integral = the integrandZf(x)dx=F(x) +cThe indefinite integral of the integrand is the indefiniteintegral
Power rule•Recall thatddx(axn) =naxn-1•Now, note that to reverse differentiation:Zxndx=xn+1n+ 1forn6=-1•Example: findRx0.5dxZx0.5dx=x0.5+10.5 + 1+c=x1.51.5=23x1.5+c•Note what happens when we differentiate this function:ddx23x1.5+c=x0.5•Also note thatRdx=R1dx=Rx0dx=x0+10+1+c=x+c
Multiplicative constant rule•Recall thatddxAx=A,ddxAf(x) =Af0(x)•To reverse thisZAf(x)dx=AZf(x)dx=AF(x) +c•Example: findR100x3dxZ100x3dx= 100Zx3dx= 100x44+c= 25x4+c•Derivative of this function is ...
Sum/difference rule•Recall thatddx(f(x)±g(x)) =f0(x)±g0(x)•To reverse thisZ[f(x)±g(x)]dx=Zf(x)dx±Zg(x)dx=F(x)±G(x)+c•Example: findR(5x3+ 3x2)dxZ(5x3+ 3x2)dx=Z5x3dx+Z3x2dx= 5Zx3dx+ 3Zx2dx+c= 5x44+ 3x33+c= 1.25x4+x3+c•Derivative of this function is ...
Function of a function rule•This rule reverses the chain rule of differentiation•Recall that the chain rule states thatddx(f(x))n=nf(x)n-1df(x)dx=nf(x)n-1
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