Evaluating the Definite Integral of \(|x|/x\)

The question asks us to evaluate the definite integral \(l = \int_a^b \frac{|x|}{x} dx\), given that \(a \lt b\) and specifically for the case where \(a \lt 0 \lt b\).

The function inside the integral is \(f(x) = \frac{|x|}{x}\). This function is defined for all \(x \neq 0\). Let's analyze its behavior:

• If \(x > 0\), then \(|x| = x\). So, \(f(x) = \frac{x}{x} = 1\).
• If \(x < 0\), then \(|x| = -x\). So, \(f(x) = \frac{-x}{x} = -1\).

Thus, the function \(f(x)\) is a piecewise constant function:

$$ \frac{|x|}{x} = \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \end{cases} $$

We are given that \(a \lt 0 \lt b\). This means the interval of integration \([a, b]\) includes both negative and positive values, crossing through \(x=0\).

To evaluate the integral \(\int_a^b \frac{|x|}{x} dx\) when \(a \lt 0 \lt b\), we need to split the integral into two parts at \(x=0\), because the definition of the integrand changes at \(x=0\).

So, we can write the integral as:

$$ l = \int_a^b \frac{|x|}{x} dx = \int_a^0 \frac{|x|}{x} dx + \int_0^b \frac{|x|}{x} dx $$

Now, let's evaluate each part separately:

1. For the integral \(\int_a^0 \frac{|x|}{x} dx\): In the interval \([a, 0)\), \(x\) is negative. Therefore, \(\frac{|x|}{x} = -1\). $$ \int_a^0 \frac{|x|}{x} dx = \int_a^0 (-1) dx $$ The antiderivative of \(-1\) is \(-x\). Evaluating the definite integral: $$ \int_a^0 (-1) dx = [-x]_a^0 = (-0) - (-a) = 0 + a = a $$
2. For the integral \(\int_0^b \frac{|x|}{x} dx\): In the interval \((0, b]\), \(x\) is positive. Therefore, \(\frac{|x|}{x} = 1\). $$ \int_0^b \frac{|x|}{x} dx = \int_0^b (1) dx $$ The antiderivative of \(1\) is \(x\). Evaluating the definite integral: $$ \int_0^b (1) dx = [x]_0^b = b - 0 = b $$

Now, adding the results of the two parts:

$$ l = \int_a^0 \frac{|x|}{x} dx + \int_0^b \frac{|x|}{x} dx = a + b $$

Thus, when \(a \lt 0 \lt b\), the value of the integral \(l\) is \(a+b\).

Comparing with Options

Let's compare our result \(a+b\) with the given options:

• Option 1: \(a + b\)
• Option 2: \(a - b\)
• Option 3: \(b - a\)
• Option 4: \(\frac{(a+b)}{2}\)

Our calculated value \(a+b\) matches Option 1.

Conclusion

Based on the evaluation of the definite integral by splitting the interval at \(x=0\), we found that \(l = a+b\) when \(a \lt 0 \lt b\).

Revision Table: Definite Integral of Absolute Value

Additional Information: Properties of Definite Integrals

Understanding definite integrals is crucial for solving such problems. Here are some key properties:

• Linearity: \(\int_a^b [cf(x) + dg(x)] dx = c\int_a^b f(x) dx + d\int_a^b g(x) dx\)
• Additivity: \(\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx\), provided \(a \lt c \lt b\). This property was used in our solution to split the integral at \(x=0\).
• Reversing Limits: \(\int_a^b f(x) dx = - \int_b^a f(x) dx\)
• Integral of Constant: \(\int_a^b c dx = c(b-a)\). This was used to evaluate \(\int_a^0 (-1) dx\) and \(\int_0^b (1) dx\).

When dealing with integrands involving absolute values or other piecewise definitions, always check if the interval of integration crosses the point where the definition changes. If it does, split the integral at that point.