Section 5.1 – Antidifferentiation:  The Indefinite Integral

 

Antidifferentiation

          If F(x) is a function, and F ‘(x) = f(x), then F(x) is the antiderivative (or indefinite integral) of f(x)

 

Let’s Play:     If  , then what might F(x) =

 

 

 

 

FAMILY of Antiderivatives

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Fundamental Property of Antiderivatives

          If F(x) is an antiderivative of the continuous function f(x), then any other antiderivative of

f(x) has the form G(x) =Ff(x) +C

 

Indefinite Integral of f(x)         

Terminology

  integrand symbol                      f(x) -  Integrand    

  C – constant of Integration          x – Variable of integration

Let’s try some using “Guess & Check”

 

                                                       

 

 

 

 

 

Rules for Integrating Common Functions

 

The Constant Rule:      

 

The Power Rule:           

 

The Logarithmic Rule:   

 

The Exponential Rule:    

 

Examples:

                                                                                                               

 

 

 

 

 

 

                                                                                                      

 

 

 

 

 

 

                                                         

 

 

 

 

 

                                                                                     

 

 

 

 

 

 

 

 

 

 

 

 

Algebraic Rules for Indefinite Integration

 

 

The Constant Multiple Rule:   

 

The Sum Rule:                            

 

The  Different Rule:                   

 

Examples

 

                                               

 

 

 

 

 

 

 

 

 

                                                            

 

 

 

 

 

 

 

 

 

 

Differential Equation     

 

Solve the given initial value problem for y = f(x)

#32      where y = 3  when x = 0

 

 

 

 

 

 

 

The slope f ‘ (x) at each point (x,y) on a curve y = f(x) is given along with a particular point (a,b) on the curve.  Use this information to find f(x).

#38   

 

 

 

 

 

 

 

 

 

Applications

#42  MARGINAL PROFIT

A manufacturer estimates marginal revenue to be  dollars per unit when the level of production is q units.  The corresponding marginal cost has been found to be 0.4q  dollars per unit.  Suppose the manufacturer’s profit is $520 when the level of production is 16 units.  What is the manufacturer’s profit when the level of production is 25 units?

 

 

 

 

 

 

 

 

 

#48  SALES

The monthly sales at an import store are currently $10,000 but are expected to be declining at the rate of

 dollars per month t months from now.  The store if profitable as long as the sales level is above $8,000 per month.

a)     Find a formula for the expected sales in t months.

 

 

 

b)     What sales figure should be expected 2 years from now?

 

 

c)     For how many months will the store remain profitable?

 

 

 

 

 

#58  CANCER THERAPY

A new medical procedure is applied to a cancerous tumor with volume 30 cm3, and t days later the value is found to be changing at the rate 

a)  Find a formula for the volume of the tumor after t days.

 

 

b)  What is the volume after 60 days?   After 120 days?

 

 

c)     For the procedure to be successful, it should take no longer than 90 days for the tumor to begin to shrink.  Based on this criterion, does the procedure succeed?