91Ó°ÊÓ

When talking about integration in calculus, especially in the context of physics, one important concept is calculating the work done by a force. In simple terms, work is the amount of energy transferred when a force moves an object over a distance.
In mathematical terms, if you have a force function \(F(x)\) that varies with the position \(x\), the work \(W\) done by this force moving an object from point \(a\) to point \(b\) is calculated using the integral:

• \( W = \int_{a}^{b} F(x) \, dx \)

In our problem, the force exerted by a stamping machine is given by \(F(x) = \frac{4x}{x^2 + 3x + 2}\). We need to calculate the work done as this force moves from \(x = 0\) cm to \(x = 0.5\) cm.
This conceptual understanding sets the stage for the next critical aspect—simplifying the force function to a more manageable form for integration.

To solve the integral of a complex rational function, partial fraction decomposition is often used. It allows us to break down a fraction into simpler parts that are easier to integrate. This method is essential when dealing with rational functions where the degree of the numerator is less than the degree of the denominator.
In our given exercise, the force function \( F(x) = \frac{4x}{x^2 + 3x + 2} \) has a denominator that can be factored into \((x+1)(x+2)\).

• The goal is to express it as \( \frac{A}{x+1} + \frac{B}{x+2} \).
• Through equating and solving: \(4x = A(x+2) + B(x+1)\), we find that \(A = 4\) and \(B = -4\).

Therefore, the function transforms into:

• \( \frac{4x}{(x+1)(x+2)} = \frac{4}{x+1} - \frac{4}{x+2} \)

This decomposition simplifies the integration process in the next steps. It breaks the problem into parts that are more straightforward to handle.

After simplifying our function using partial fraction decomposition, we are ready to perform definite integration. This process not only involves finding the integral but also evaluating it over a specific interval to find a numerical result. Definite integrals are evaluated from a point \(a\) to a point \(b\), providing a specific value representing the accumulated quantity, such as total work done.

• For the decomposed function \( \frac{4}{x+1} - \frac{4}{x+2} \), the integration from \(0\) to \(0.5\) becomes:

\[ W = \int_{0}^{0.5} \left( \frac{4}{x+1} - \frac{4}{x+2} \right) \, dx = 4 \left[ \ln|x+1| - \ln|x+2| \right]_0^{0.5} \]Substituting the bounds, we find:

• \[ W = 4 \left( \ln\left( \frac{3}{5} \right) \right) \]
• The result simplifies to \[ W = 4 \ln(0.6) \approx -2.04 \text{ Joules}\]

This outcome indicates how much work is done over the given distance, and the result's negative sign suggests that the work is done against the force's direction. Understanding definite integration in this context is crucial for translating math into physical reality.