Lesson #109 Area Between Two Curves (w.r.t. y-axis)


Quote of the Day:
"I have no particular talent. I am only inquisitive."  
    -- Albert Einstein

Objectives:
The student will compute the area between 2 curves with respect
   to the y-axis as well as with respect to the x-axis.

1. Collect homework.  

2. Notice that you do not have to concern yourself about
   where the curves go below the x-axis in finding the area 
   between two curves.  The negatives work themselves out.
   Recall that in determining the total area between a 
   curve and the x-axis over a particular interval, you had 
   to find out if the curve intersected the x-axis between 
   the beginning and end of the interval.  You do not need 
   to do that here.  Look at the example below:

       

   Let the area of each region in the diagram be denoted by 
   the capital letters, A, B, C, D, and E. To determine the 
   shaded area between the two curves, you would integrate 
   f(x) from x = a to x = b and then subtract the integral 
   of g(x) over the same interval.

       

   Note that B + C + D is the area we were seeking.

3. Example:

       

    First, determine the points of intersection and draw a 
    picture of the two relations.  What do you notice about 
    this drawing (that makes it different from yesterday's 
    problem)?
 
    If you try to integrate with respect to the x-axis, you 
    will not find the total area.  You must turn integrate 
    with respect to the y-axis; in other words, the 
    rectangles that you are summing are drawn horizontally.

       

    Figure 1 shows the area that we are trying to find.
    Figure 2 shows the area that we would compute if you
             evaluated the integral with respect to the
             x-axis.
    Figure 3 shows that we are summing up rectangles which 
             are drawn horizontally - not vertically.
    Figure 4 shows the area "under" the parabola (the blue 
             and yellow shading) and the area "under" the 
             line (just the blue shaded region).

       

4. Here are the two formulas for finding areas between two curves:

       


   Emphasize that you are summing areas of rectangles
   whose base is either dy or dx, and whose height is the 
   difference between the two functions or relations.

       

5. Examples
       

6. Assignment:
   p. 448 (3, 5b, 6, 11, 13, 14, 18, 21, 31)
Lesson #109 Area Between Two Curves (w.r.t. y-axis)

Click here to go to the next page
