5.3.2 Applications of Integration

What is the constant of integration? 

• When finding an anti-derivative there is a constant term to consider
  • this constant term, usually called , is the constant of integration
• In terms of graphing an anti-derivative, there are endless possibilities
  • collectively these may be referred to as the family of antiderivatives or family of curves
  • the constant of integration is determined by the exact location of the curve
    • if a point on the curve is known, the constant of integration can be found

How do I find the constant of integration?

• For , the constant of integration,  - and so the particular antiderivative - can be found if a point the graph of  passes through is known 

 If need be, rewrite into an integrable form

 Each term needs to be a power of  (or a constant)

 Integrate each term of, remembering the constant of integration, “”

Substitute the  and  coordinates of a given point in to  to form an equation in 

Solve the equation to find 

What is meant by the area under a curve?

• The phrase “area under a curve” refers to the area bounded by
  • the graph of 
  • the -axis
  • the vertical line 
  • the vertical line 
• The exact area under a curve is found by evaluating a definite integral
• The graph of  could be a straight line
  • the use of integration described below would still apply
    • but the shape created would be a trapezoid
    • so it is easier to use “”

What is a definite integral?

• This is known as the Fundamental Theorem of Calculus
• a and b are called limits
  • a is the lower limit
  • b is the upper limit
•  is the integrand
•  is an antiderivative of
• The constant of integration (“”) is not needed in definite integration
  • "” would appear alongside both F(a) and F(b)
  • subtracting means the “”’s cancel

How do I form a definite integral to find the area under a curve?

• The graph of  and the -axis should be obvious boundaries for the area so the key here is in finding  and  - the lower and upper limits of the integral 

Look out for the -axis being one of the (vertical) boudnaries – in this case the limit () will be 0

One, or both, of the limits, could be a root of the equation

i.e.  where the graph of crosses the -axis

In this case solve the equation to find the limit(s)

A GDC will solve this equation, either from the graphing screen or the equation solver

Does my calculator/GDC do definite integrals?

• Modern graphic calculators (and some ‘advanced’ scientific calculators) have the functionality to evaluate definite integrals
  • i.e. they can calculate the area under a curve (see above)
• If a calculator has a button for evalutaing definite integrals it will look something like

• This may be a physical button or accessed via an on-screen menu
• Some GDCs may have the ability to find the area under a curve from the graphing screen
• Be careful with any calculator/GDC, they may not produce an exact answer

How do I use my GDC to find definite integrals?

Without graphing first …

• Once you know the definite integral function your calculator will need three things in order to evaluate it
  • The function to be integrated (integrand) ()
  • The lower limit ( from )
  • The upper limit ( from )
• Have a play with the order in which your calculator expects these to be entered – some do not always work left to right as it appears on screen!

With graphing first ...

• Plot the graph of 
  • You may also wish to plot the vertical lines  and 
    • make sure your GDC is expecting an "" style equation
  • Once you have plotted the graph you need to look for an option regarding “area” or a physical button
    • it may appear as the integral symbol (e.g. )
    • your GDC may allow you to select the lower and upper limits by moving a cursor along the curve - however this may not be very accurate
    • your GDC may allow you to type the exact limits required from the keypad
      • the lower limit would be typed in first
      • read any information that appears on screen carefully to make sure