$\"\"$

The curve is $\"\"$ , $\"\"$ and $\"\"$ .

Length of the curve: $\"\"$ .

Here $\"\"$ and $\"\"$ .

$\"\"$ .

Differentiate on each side with respect to $\"\"$ .

$\"\"$

Simpson $\"\"$ s rule:

Let $\"\"$ be continuous on $\"\"$ let $\"image\"$ be an even integer,

The Simpson $\"\"$ s Rule for approximating $\"image\"$ is given by

$\"\"$

where $\"image\"$ and $\"\"$ .

$\"\"$

Here $\"\"$ .

Substitute corresponding values in simpson $\"\"$ s formula.

$\"\"$

Consider a table for function values corresponding $\"\"$ values.

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Substitute the results in the table in $\"\"$ .

$\"\"$

$\"\"$ .

Arc length of the curve is $\"\"$ .

$\"\"$

By using calculator we will get the value of

$\"\"$ .

Above value is very closer to the Simpson $\"\"$ s approximation.

$\"\"$

Using simpson $\"\"$ s rule, arc length of the curve $\"\"$ is $\"\"$ .

Using calculator, arc length of the curve $\"\"$ is $\"\"$ .