find the total area bounded by the x - axis and the curve y = f(x) on the indicated interval. f(x)=x² - x…

Explanation:

Step1: Find the integral of the function

The area (A) between the curve (y = f(x)) and the (x -)axis on the interval ([a,b]) is given by (A=\int_{a}^{b}|f(x)|dx). First, find the integral of (y = x^{2}-x - 12), (\int(x^{2}-x - 12)dx=\frac{x^{3}}{3}-\frac{x^{2}}{2}-12x + C).

Step2: Evaluate the definite - integral

Evaluate (\int_{3}^{6}(x^{2}-x - 12)dx=\left[\frac{x^{3}}{3}-\frac{x^{2}}{2}-12x\right]_{3}^{6}). [ \begin{align*} &\left(\frac{6^{3}}{3}-\frac{6^{2}}{2}-12\times6\right)-\left(\frac{3^{3}}{3}-\frac{3^{2}}{2}-12\times3\right)\ =&\left(\frac{216}{3}-\frac{36}{2}-72\right)-\left(\frac{27}{3}-\frac{9}{2}-36\right)\ =&(72 - 18-72)-\left(9-\frac{9}{2}-36\right)\ =& - 18-\left(9-\frac{9}{2}-36\right)\ =&-18-(9 - 4.5-36)\ =&-18-(-31.5)\ =&13.5 \end{align*} ]

Answer:

(13.5)