Question

The value of \(\int_0^3 |3x^2 - 3| dx\) is equal to

• A. 16
• B. 18
• C. 20
• D. 22
• E. 24

Hint:

Always visualize the parabola \(y = 3x^2-3\). It dips below the x-axis between its roots. The area below the axis must be subtracted (negated) to get the absolute value sum.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To integrate an absolute value function, we must split the integral at the points where the expression inside the modulus changes its sign.

Step 2: Key Formula or Approach:
Identify roots of \(3x^2 - 3 = 0\):
\(3(x - 1)(x + 1) = 0 \implies x = 1\) (since -1 is outside the range [0, 3]).
Intervals: [0, 1] and [1, 3].

Step 3: Detailed Explanation:
1. In [0, 1]: \(3x^2 - 3 \leq 0\), so \(|3x^2 - 3| = 3 - 3x^2\).
2. In [1, 3]: \(3x^2 - 3 \geq 0\), so \(|3x^2 - 3| = 3x^2 - 3\).
Total Integral:
\[ I = \int_0^1 (3 - 3x^2) dx + \int_1^3 (3x^2 - 3) dx \]
Calculate the first part:
\[ [3x - x^3]_0^1 = (3 - 1) - (0) = 2 \]
Calculate the second part:
\[ [x^3 - 3x]_1^3 = (27 - 9) - (1 - 3) = 18 - (-2) = 20 \]
Sum: \(I = 2 + 20 = 22\).

Step 4: Final Answer:
The value of the integral is 22.