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 xaxis 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 yaxis 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 xaxis  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 

Step 4:  Note the limits between which your slice moves  for example here the two points of intersection are 

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. 