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 (x) - axis and the curve (y = f(x)) on the interval ([a,b]) is given by (A=\int_{a}^{b}|f(x)|dx). Since (f(x)=6 + 6e^{0.2x}>0) for all real (x), then (A=\int_{2}^{5}(6 + 6e^{0.2x})dx).

Step2: Use integral rules

We know that (\int_{2}^{5}(6 + 6e^{0.2x})dx=\int_{2}^{5}6dx+\int_{2}^{5}6e^{0.2x}dx). For (\int_{2}^{5}6dx), using the rule (\int kdx=kx + C) ((k) is a constant), we have (\int_{2}^{5}6dx=6x\big|{2}^{5}=6(5 - 2)=18). For (\int{2}^{5}6e^{0.2x}dx), let (u = 0.2x), then (du=0.2dx) and (dx = 5du). When (x = 2), (u=0.2\times2 = 0.4); when (x = 5), (u=0.2\times5 = 1). So (\int_{2}^{5}6e^{0.2x}dx=6\times5\int_{0.4}^{1}e^{u}du=30e^{u}\big|_{0.4}^{1}=30(e^{1}-e^{0.4})).

Step3: Calculate the total area

(A = 18+30(e - e^{0.4})). [ \begin{align*} A&=18+30(e - e^{0.4})\ &=18+30(2.71828 - 1.49182)\ &=18+30\times1.22646\ &=18 + 36.7938\ &=54.7938\approx54.794 \end{align*} ]

Answer:

(54.794)