Download Section 6.2

AP CALCULUS NOTES
SECTION 6.2 THE INDEFINITE INTEGRAL; INTEGRAL CURVES AND SLOPE
FIELDS
Some situations arise where you are asked to find a function F whose derivative is a known function
f. For example, an engineer who can measure the variable rate at which water is leaking from a tank
might want to know the total amount leaked over a certain period of time. Also, a biologist who
knows the rate at which a bacteria population is increasing might want to know the size of the
population in the near future. If such a function F exists,
1.) A function F is called an antiderivative of f on an interval I if F   x   f  x  for all x in I.
Ex.1.) f  x   x4
Possible Antiderivatives:
1
F  x   x5  3
5
1
F  x   x5  1
5
1
F  x   x5
5
1 5
F  x  x  2
5
Ex.2.) f  x   cos x
Possible Antiderivatives:
F  x   sin x  4
F  x   sin x  2
F  x   sin x
F  x   sin x  3
THEOREM: If F is an antiderivative of f on an interval I, then the most general antiderivative of
f on I is F  x   C where C is an arbitrary constant. Thus, the most general antiderivative of f on I
is a family of functions F  x   C whose graphs are vertical translations of one another.
A.) The Indefinite Integral: used to denote the collection of all antiderivatives of a function f.
Indefinite Integral of f with respect to x:  f  x  dx
Integration – process of finding antiderivatives.
d
 F  x    f  x 
 f  x  dx  F  x   C
dx 
Find the general antiderivative of each of the following functions: (Table on pg. 380)
Ex.3.) f  x   3x  2
Ex.4.) f  x  
3
 x
 cos  
x
2
Ex.5.) f  x  
1 2x
e
x
Ex.6.) f  x   4sin x 
2 x5  x
x
Antiderivative Linearity Rules:
1.) Constant Multiple Rule:
Function
kf  x 
General Antiderivative
kF  x   C, k is a constant
2.) Negative Rule:
 f  x
F  x   C
3.) Sum or Difference Rule:
f  x  g  x
F  x  G  x  C
In calculus, oftentimes you have to find a function given knowledge about its derivative
B.) Differential Equations – an equation that involves a derivative of an unknown function.
dy
 f  x  is called a differential equation
dx
The combination of a differential equation and an initial condition is called an initial value
problem. By solving an initial value problem, we find the particular solution for y as opposed to
the general solution y  F  x   C .
Ex.7.) Find f if f   x   12x2  6x  4;
f 1  1 .
Ex.8.) Solve the initial value problem given
dy
20
 ex 
and y  0  2 .
dx
1  x2
Ex.9.) A balloon ascending at a rate of 12 ft/s is at a height 80 ft above the ground when a package
is dropped from the balloon. Given the acceleration of gravity is 32 ft/s 2 , how long does it take the
package to reach the ground?
C.) Slope Fields (The Geometry of Antiderivatives) – provide a visual of a family of
antiderivatives (solutions of differential equations)
 Graphs of antiderivatives of a function f are called integral curves. All integral curves of
a function f create a slope field.
dy
 2x .
Ex.10.) Draw a slope field for the differential equation
dx
dy
dy
dy
 x, y 
 x, y 
 x, y 
dx
dx
dx
 3, 2
 1, 0
1, 2
 3, 1
 3, 0 
 3,1
 3, 2 
 2, 2
 2, 1
 2,0
 2,1
 2, 2
 1, 2
 1, 1
 1,1
 1, 2 
 0, 2
 0, 1
 0, 0 
 0,1
 0, 2 
1, 2
1, 1
1,0
1,1
 2, 2
 2, 1
 2, 0 
 2,1
 2, 2 
 3, 2 
3, 1
3,0
 3,1
3, 2
Find the particular solution y  f  x 
to the given differential equation with
the initial condition f  0  1 .
Graph this particular integral curve on
the slope field above.
Note: all of the points that have the same x-coordinate will have the same slope since our
differential equation has an x-term but no y-term.
Ex.11.) Match each slope field with the differential equation each slope field could represent.
A.)
B.)
C.)
D.)
dy
 x2
dx
dy 2

2.)
dx x
dy
 cos x
3.)
dx
1.)
dy
 sin x
dx
dy
 ex
5.)
dx
dy 1 3
 x
6.)
dx 4
4.)