Area for $y \le 0$

Calculus Level pending

What is the area of the region bounded by $y \geq x^3 + 6x^2 + 9x \text{ and } y \leq 0$ in the domain $-3 \leq x \leq 0?$

\frac{27}{16}

\frac{27}{2}

\frac{27}{4}

\frac{27}{8}

1 solution

Tom Engelsman
Nov 8, 2020

The area is computed per:

$A = \int_{-3}^{0} 0 - (x^3+6x^2+9x) dx = -\frac{x^4}{4} - 2x^3 -\frac{9x^2}{2}|_{-3}^{0} = \boxed{\frac{27}{4}}.$