Question

The integral \( \int_{-2}^4 x^2 |x| \, dx \) is equal to:

• A. 72
• B. 68
• C. 64
• D. 48
• E. 37

Hint:

When integrating absolute values, break the integral at points where the function inside the absolute value changes sign.

Answer 1

The Correct Option is B

Step 1: Split the integral into two parts based on the absolute value function: \[ \int_{-2}^4 x^2 |x| \, dx = \int_{-2}^0 x^2 (-x) \, dx + \int_0^4 x^2 x \, dx. \] Step 2: Simplify each integral: \[ \int_{-2}^0 -x^3 \, dx + \int_0^4 x^3 \, dx. \] Step 3: Now, compute each integral: \[ \int_{-2}^0 -x^3 \, dx = \left[ -\frac{x^4}{4} \right]_{-2}^0 = -\left(0 - \frac{16}{4}\right) = 4, \] \[ \int_0^4 x^3 \, dx = \left[ \frac{x^4}{4} \right]_0^4 = \frac{256}{4} - 0 = 64. \] Step 4: Adding these results gives: \[ 4 + 64 = 68. \]