Transcript Business Calculus

Business Calculus
Improper Integrals,
Differential Equations
 5.3 Improper Integrals
Consider an integral to represent the area under the curve shown
from x = 0 to x = ∞ :
Question: could such an area be finite?
b
Recall the Basic Area Integral =
 top curve  bottom curvedx
a
To determine the area (whether finite or infinite), we set up an
integral. Since the limits of integration in our example are not
both finite numbers, we call this an improper integral.

For our example, area =

2e  0.4 x dx
0
Since ∞ is not a number, we cannot simply use the first
fundamental theorem of calculus. We must rewrite the integral
to become a definite integral.
Create a definite integral by arbitrarily stopping the right side of
the interval at some finite, but not designated value, b.
This creates a definite integral, which can be evaluated.
Then, to get the full area from 0 to ∞, we allow b to ‘slide’ to ∞
using a limit.

For our example: Area =

0
b

2e  0.4 x dx = lim 2e  0.4 x dx
b 
0
To evaluate this area, we work out the definite integral first,
copying the limit at each step.
When the definite integral is evaluated, we take the limit as b
approaches infinity.
If this limit yields a finite number, then the area is finite, and we
say the improper integral converges.
If this limit gives an infinite result, then the area is infinite,
and we say the integral diverges.
 5.7 Differential Equations
A differential equation is an equation which involves a derivative.
examples of differential equations:
dy
 x2  5
dx
dy
 xy
dx
dP
 kP
dt
 xD( x)
E ( x) 
D( x )
To solve a differential equation means to find the original function
whose derivative appears in the equation.
Steps to solve a differential equation:
I. Isolate the derivative, if necessary.
II. If the derivative is equal to a formula involving the input
variable only, then the original function (solution) is just the
antiderivative.
III. If the derivative is equal to a formula involving the output
variable, then the variables must be separated. This is called
the separation of variables technique.
IV. If additional information is given, solve for the constant of
integration.
Note: variables can only be separated using multiplication or
division, never addition or subtraction.
 Separation of Variables
dy
Example: solve
 xy
dx
Note: the derivative is isolated, and is in the correct form.
Multiply both sides by dx.
dy  xy dx
Separation means leave any x variables with dx, and move any
y variables to the dy side of the equation.
1
Divide both sides by y.
dy  xdx
y
Next, integrate both sides of the equation. Be careful to notice
the variable of the integral in each case.


1
1 2
dy  xdx  ln y  x  c
y
2
If possible, solve for the output variable, in this case y.
e
ln y
e
ye
1
2
1
2
x2
y  Ce
1
2
x 2 c
e
x2
c