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{1}{7x - 1}>0) for (x\in[4,5]), then (A = \int_{4}^{5}\frac{1}{7x - 1}dx).

Step2: Use substitution

Let (u = 7x-1), then (du=7dx). When (x = 4), (u=7\times4 - 1=27), and when (x = 5), (u=7\times5 - 1 = 34). So (\int_{4}^{5}\frac{1}{7x - 1}dx=\frac{1}{7}\int_{27}^{34}\frac{1}{u}du).

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

We know that (\int\frac{1}{u}du=\ln|u|+C). So (\frac{1}{7}\int_{27}^{34}\frac{1}{u}du=\frac{1}{7}[\ln(u)]_{27}^{34}).

Step4: Evaluate the definite - integral

(\frac{1}{7}[\ln(u)]_{27}^{34}=\frac{1}{7}(\ln(34)-\ln(27))=\frac{1}{7}\ln(\frac{34}{27})\approx\frac{1}{7}\times0.231\approx0.033).

Answer:

(\frac{1}{7}\ln(\frac{34}{27})\approx0.033)