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) bounded by 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}{\sqrt{8x + 5}}>0) for (x\in[6,8]), then (A=\int_{6}^{8}\frac{8}{\sqrt{8x + 5}}dx).

Step2: Use substitution

Let (u = 8x+5), then (du=8dx). When (x = 6), (u=8\times6 + 5=53); when (x = 8), (u=8\times8+5 = 69). The integral becomes (\int_{53}^{69}\frac{du}{\sqrt{u}}).

Step3: Integrate

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

Step4: Evaluate the definite integral

(\left[2\sqrt{u}\right]_{53}^{69}=2\sqrt{69}-2\sqrt{53}\approx2(8.307 - 7.280)=2\times1.027 = 2.054).

Answer:

(2\sqrt{69}-2\sqrt{53}\approx2.054)