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)=\frac{8}{3x + 6}>0) for (x\in[4,7]), then (A=\int_{4}^{7}\frac{8}{3x + 6}dx).

Step2: Use substitution

Let (u = 3x+6), then (du=3dx) and (dx=\frac{1}{3}du). When (x = 4), (u=3\times4 + 6=18), when (x = 7), (u=3\times7+6 = 27). The integral becomes (\int_{18}^{27}\frac{8}{u}\times\frac{1}{3}du=\frac{8}{3}\int_{18}^{27}\frac{1}{u}du).

Step3: Integrate (\frac{1}{u})

We know that (\int\frac{1}{u}du=\ln|u|+C). So (\frac{8}{3}\int_{18}^{27}\frac{1}{u}du=\frac{8}{3}[\ln(u)]_{18}^{27}).

Step4: Evaluate the definite - integral

(\frac{8}{3}(\ln(27)-\ln(18))=\frac{8}{3}\ln(\frac{27}{18})=\frac{8}{3}\ln(\frac{3}{2})\approx1.099).

Answer:

(\frac{8}{3}\ln(\frac{3}{2})\approx1.099)