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Approximate the area between the curve of
*y* = *x*
^{2}
and the
*x* - *axis*
from
*x* = 0
to
*x* = 1
:

Figure %: Area bounded by a curve

The method to be used in this section to solve this problem is Riemann sums, which involves subdividing the region into rectangles of equal width and adding up the areas of all of the rectangles to approximate the area of the region.

Let's first use three subdivisions to approximate this area:

Figure %: Three subdivision of the region

Each of the regions has a width of 1/3 the interval from
*x* = 0
to
*x* = 1
which is 1/3. These will form the bases of the
rectangles, but what should be chosen for the height? One possibility would be to use the value of the function at the left
endpoint of each subdivision as the height. This is called a left-hand approximation.

Figure %: Left-hand approximation of area using three subdivisions

The left-hand approximation for the area is as follows:

(0) + + =

Another possibility is to use the value of the function at the right endpoint of each subdivision as the height. This is called a right-hand approximation.

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