1. 2. Area Between Curves

2. Area
Up to now, we’ve only considered area between a curve and the x-axis.
We now look at a way to find the area of a region bounded by 2 or more curves.

3. Area
We first learned to approximate areas by using rectangular approximations. You can think of these rectangles as being slices of the region. The sum of the areas of the slices gives you an approximation of the area of the region.
If we can find a way to represent the height of each of these representative rectangular slices, we can integrate through a uniformly, infinitely small width.

4. Area
Here is what the slices might look like between two curves f and g on an interval [a,b]
The height of a representative rectangle is f(x) – g(x) or TOP – BOTTOM
Therefore the area is

5. Horizontal slices
Sometimes because of the shape of the function, it is better to use horizontal slices.
In this case, we would integrate with respect to y and use RIGHT - LEFT

6. Steps Draw a picture Clearly identify region
Decide how to slice the region – vertically or horizontally
Draw a representative rectangle (line)
Find/identify intervals of integration
Set up the equation for the height of a representative rectangle

7. Example 1
Find the area of the region bounded by and the vertical lines x=0 and x=1

8. Example 2
Find the area of the region bounded by

9. Example 3
Find the area of the region bounded by

10. Example 4
Find the area of the region bounded by

11. Example 5 - split into subregions
Find the area enclosed by above and the x-axis and y=x-2 below

12. Example 6 – horizontal slices
Find the area enclosed by above and the x-axis and y=x-2 below

13. Example 7
Find the area enclosed by and

14. Representative elements
In this section we found area using a known geometric formula (A=hw), a representative element (slice) and integration. We will continue with this concept in the next section.
Here is what we did today: