Question

The value of $\int_{-1}^{2} 4x^2 |x|\, dx$ is equal to:

• A. $17$
• B. $16$
• C. $15$
• D. $14$
• E. $13$

Hint:

When integrating absolute values, visualising the function as two separate curves joined at the origin helps ensure you apply the correct signs to the power functions.

Answer 1

The Correct Option is A

Concept: To integrate a function containing an absolute value $|x|$, we must split the integral at the point where the expression inside the absolute value changes sign (at $x=0$).
• For $x < 0$, $|x| = -x$.
• For $x \geq 0$, $|x| = x$.

Step 1: Split the integral into two parts.
\[ \int_{-1}^{2} 4x^2 |x| \, dx = \int_{-1}^{0} 4x^2 (-x) \, dx + \int_{0}^{2} 4x^2 (x) \, dx \] \[ = \int_{-1}^{0} -4x^3 \, dx + \int_{0}^{2} 4x^3 \, dx \]

Step 2: Perform the integration for each part.
For the first part: \[ \left[ -4 \cdot \frac{x^4}{4} \right]_{-1}^{0} = [ -x^4 ]_{-1}^{0} = -(0)^4 - (-(-1)^4) = 0 - (-1) = 1 \] For the second part: \[ \left[ 4 \cdot \frac{x^4}{4} \right]_{0}^{2} = [ x^4 ]_{0}^{2} = (2)^4 - (0)^4 = 16 - 0 = 16 \]

Step 3: Sum the results.
\[ \text{Total Value} = 1 + 16 = 17 \]