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

Explanation:

Step1: Recall area - integral formula

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). Since (f(x)=6x^{2}+2>0) for all real (x) (because (x^{2}\geq0) for all (x\in R), so (6x^{2}+2\geq2>0)), then (|f(x)| = f(x)) and (A=\int_{-5}^{-2}(6x^{2}+2)dx).

Step2: Apply integral rules

We know that (\int(6x^{2}+2)dx=6\int x^{2}dx + 2\int dx). Using the power - rule for integration (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have (6\times\frac{x^{3}}{3}+2x=2x^{3}+2x+C).

Step3: Evaluate the definite integral

(\int_{-5}^{-2}(6x^{2}+2)dx=\left[2x^{3}+2x\right]_{-5}^{-2}). [ \begin{align*} &(2\times(-2)^{3}+2\times(-2))-(2\times(-5)^{3}+2\times(-5))\ =&(2\times(-8)-4)-(2\times(-125)-10)\ =&(-16 - 4)-(-250-10)\ =&-20 + 260\ =&240 \end{align*} ]

Answer:

240