12. The figure on the right shows the graph of \( f(x) = x^3 - x^2 - 6x \). Find the area between the graph of \( f(x) \) and the x-axis on \( [-2, 3] \).






Answer is: option2

\( \frac{253}{12} \)

Solution:

Area = \[ \int_{-2}^3 \left| x^3 - x^2 - 6x \right| \, dx = \left| \left[ \frac{x^4}{4} - \frac{x^3}{3} - 3x^2 \right]_{-2}^0 \right| + \left[ \frac{x^4}{4} - \frac{x^3}{3} - 3x^2 \right]_0^3 \]

\[ = \left| \frac{16}{3} \right| + \left| \frac{-63}{4} \right| = \frac{16}{3} + \frac{63}{4} = \frac{253}{12} \]

So, the correct answer is (B) \( \frac{253}{12} \).

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