Question

The value of \(\int^{2}_{-1}(x-2)|x|)dx\) is equal to

• A. \(\frac{-1}{2}\)
• B. \(\frac{-3}{2}\)
• C. \(\frac{-5}{2}\)
• D. \(\frac{-7}{2}\)
• E. \(\frac{-9}{2}\)

Answer 1

The Correct Option is D

We are tasked with evaluating the integral: \[ I = \int_{-1}^{2} (x - 2 |x|) \, dx. \]
Step 1: Break the absolute value function into cases The function \( |x| \) can be written as: \[ |x| = \begin{cases} -x & \text{for} \, x < 0, \\ x & \text{for} \, x \geq 0. \end{cases} \] Therefore, we need to split the integral into two parts, one for \( x \in [-1, 0] \) and the other for \( x \in [0, 2] \).
Step 2: Split the integral into two parts For \( x \in [-1, 0] \), \( |x| = -x \). Hence, the integrand becomes: \[ x - 2|x| = x + 2x = 3x. \] For \( x \in [0, 2] \), \( |x| = x \). Hence, the integrand becomes: \[ x - 2|x| = x - 2x = -x. \] Thus, we can split the integral into two parts: \[ I = \int_{-1}^{0} 3x \, dx + \int_{0}^{2} -x \, dx. \]
Step 3: Evaluate each integral Integral 1: \( \int_{-1}^{0} 3x \, dx \) \[ \int_{-1}^{0} 3x \, dx = \frac{3}{2} x^2 \Bigg|_{-1}^{0} = \frac{3}{2} (0^2 - (-1)^2) = \frac{3}{2} (0 - 1) = -\frac{3}{2}. \]
Integral 2: \( \int_{0}^{2} -x \, dx \) \[ \int_{0}^{2} -x \, dx = -\frac{1}{2} x^2 \Bigg|_{0}^{2} = -\frac{1}{2} (2^2 - 0^2) = -\frac{1}{2} (4) = -2. \]
Step 4: Combine the results Now, combine both parts of the integral: \[ I = -\frac{3}{2} + (-2) = -\frac{3}{2} - \frac{4}{2} = -\frac{7}{2}. \]

The correct option is (D) : \(\frac{-7}{2}\)

Answer 2

Evaluating the Integral 

We are tasked with evaluating the integral: I = ∫_-1² (x - 2 |x|) dx

Step 1: Break the absolute value function into cases

The function |x| can be written as:

|x| =    {        -x  for x < 0,        x   for x ≥ 0.    }Therefore, we need to split the integral into two parts, one for \( x \in [-1, 0] \) and the other for \( x \in [0, 2] \).

 

Step 2: Split the integral into two parts

For \( x \in [-1, 0] \), \( |x| = -x \). Hence, the integrand becomes: x - 2|x| = x + 2x = 3x.

For \( x \in [0, 2] \), \( |x| = x \). Hence, the integrand becomes: x - 2|x| = x - 2x = -x.

Thus, we can split the integral into two parts: I = ∫_-1⁰ 3x dx + ∫₀² -x dx .

Step 3: Evaluate each integral

Integral 1: ∫_-1⁰ 3x dx

 

    ∫_-1⁰ 3x dx = (3/2) x² |_-1⁰    = (3/2) (0² - (-1)²)    = (3/2) (0 - 1)    = -3/2.    

 

Integral 2: ∫₀² -x dx

 

    ∫₀² -x dx = -(1/2) x² |₀²    = -(1/2) (2² - 0²)    = -(1/2) (4)    = -2.    

 

Step 4: Combine the results

Now, combine both parts of the integral: I = -3/2 + (-2) = -3/2 - 4/2 = -7/2.

The correct option is (D): \(\frac{-7}{2}\)