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Step-by-step Solution
Step 1: Identify the Given Restoring Force
The force required to stretch the rubber band is given by
$F = a\,x + b\,x^2$, where $a$ and $b$ are constants.
Step 2: Recall the Formula for Work Done
The small amount of work done $dW$ in stretching the band a small distance $dx$ is:
$dW = F \, dx = (a\,x + b\,x^2)\,dx$.
Step 3: Integrate Over the Required Limits
To find the total work $W$ done when the band is stretched from $x=0$ to $x=L$, we integrate:
$W = \int_{0}^{L} (a\,x + b\,x^2)\,dx$
Step 4: Perform the Integration
Splitting the integral and integrating term-by-term gives:
$W = \int_{0}^{L} a\,x \,dx + \int_{0}^{L} b\,x^2 \,dx
= a \int_{0}^{L} x \,dx + b \int_{0}^{L} x^2 \,dx.
$
$W = a \left[\frac{x^2}{2}\right]_{0}^{L} + b \left[\frac{x^3}{3}\right]_{0}^{L}.
$
Step 5: Evaluate the Result
Substituting the limits $0$ and $L$:
$W = a \left(\frac{L^2}{2}\right) + b \left(\frac{L^3}{3}\right)
= \frac{a\,L^2}{2} + \frac{b\,L^3}{3}.
Final Answer
The work done in stretching the rubber band is
$ \frac{a\,L^2}{2} + \frac{b\,L^3}{3}. $
