$\"\"$

\

The displacement of the spring is $\"\"$ .

\

Find the average value of $\"\"$ on the interval $\"\"$ to $\"\"$ .

\

Average value of the function $\"\"$ on $\"\"$ is defined as $\"\"$ .

\

In this case $\"\"$ and $\"\"$ .

\

The average value of $\"\"$ on the interval $\"\"$ to $\"\"$ is $\"\"$ .

\

$\"\"$ .

\

Consider $\"\"$ .

\

Solve the integral using integration by parts.

\

Formula for integration by parts : $\"\"$ .

\

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

\

Consider $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

Apply integral on each side.

\

$\"\"$

\

$\"\"$ .

\

Substitute corresponding values in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Again apply integration by parts.

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Consider $\"\"$ .

\

Solve the integral using integration by parts.

\

Formula for integration by parts: $\"\"$ .

\

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

\

Consider $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

Apply integral on each side.

\

$\"\"$

\

$\"\"$ .

\

Substitute corresponding values in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Again apply integration by parts.

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Substitute $\"\"$ and $\"\"$ in

\

$\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

The average value is $\"\"$ .

\

$\"\"$

\

The average value is $\"\"$ .