2. Area between curves
Area
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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.
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.
Here is what the slices might look like between two curves f and g on an interval [a,b]
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The height of a representative rectangle is f(x) – g(x) or TOP – BOTTOM
b
Therefore the area is
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A   ( f ( x)  g ( x))dx
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Steps
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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
Draw a picture
Clearly identify region
Decide how to slice the region – vertically or horizontally
Draw a representative rectangle
Find/identify intervals of integration
Set up the equation for the height of a representative rectangle
Example 1
Find the area of the region bounded by
f ( x)  e x , g ( x)  x, and the vertical lines x=0 and x=1
Example 2
Find the area of the region bounded by
y  x 2 , and y  2 x  x 2
Example 3
Find the area of the region bounded by
f ( x) 
Example 4
Find the area of the region bounded by
f ( x)  3x3  x 2  10 x and g ( x)  2 x  x 2
Example 5 – split into subregions
Find the area enclosed by y  x
Example 6 – horizontal slices
Find the area enclosed by y 
Example 7
Find the area enclosed by
x
x 1
2
and g ( x)  x 4  x
above and the x-axis and y=x-2 below
x above and the x-axis and y=x-2 below
y  x3 and x  y 2  2
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