Dr. J's Maths.com
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are explained in simple terms.

Calculus - Integration - Finding areas.
Summary of the steps.


 

There are several types of question focused on finding area using calculus.

The most common formats for the functions we encounter - are listed below
together with the steps to follow to obtain the require area:

1. Basic area between the curve for a function and the x axis: Step 1: Draw a rough sketch of the function.
Step 2: Add a thin slice from the x-axis to the curve (i.e. the y distance).
Step 3:

Write the integral statement as

to summarise the area of the slice.
  Step 4:

Include the limits between which your slice moves - for example
  Step 5: Replace y with the function in x.
  Step 6: Evaluate the integral in x using the appropriate technique.
  Step 7: Report your result.
     
2. Basic area between the curve for a function and the y axis: Step 1: Draw a rough sketch of the function.
Step 2: Add a thin slice from the y-axis to the curve
(i.e. the x distance).
Step 3:

Write the integral statement as

to summarise the area of the slice.
  Step 4:

Note the limits between which your slice moves (here in the vertical direction) - for example
  Step 5: Replace x with the function in y.
  Step 6: Evaluate the integral in y using the appropriate technique.
  Step 7: Report your result.
     
3. Area between 2 curves with one point of intersection. Step 1: Draw a rough sketch of the two functions.
Calculate the x value for their point of intersection (if not given).
Step 2: Add two thin slices from the x-axis - one to each of the curves (i.e. the y distance).
Steps 3 - 6:

Follow steps 3 to 6 above for type 1 for each of the two areas separately.
  Step 7: Add the two separate areas together.
     
Area between 2 curves with two points of intersection. Step 1: Draw a rough sketch of the two functions.
Calculate the x values for their points of intersection (if not given).
Step 2: Add a thin slice starting from the top curve and finishing at the lower curve
(i.e. the y distance between the two functions).
Step 3:

Write the integral statement as

to summarise the area of the slice with y being the distance between the curves
(measured by Top curve minus Bottom curve).
  Step 4:

Note the limits between which your slice moves - for example here the two points of intersection are
x = 0 and x = 2.
  Step 5:

Replace y (or f(x) and g(x)) with the difference between the functions in terms of x.

NOTE: It does not matter if one of your curves goes below the x axis. The difference approach eliminates any problems.
  Step 6: Evaluate the integral in x using the appropriate technique.
  Step 7: Report your result.