Key Concepts and Formulas

• Area Between Curves: If $f(x) \ge g(x)$ on the interval $[a, b]$, the area between the curves $y = f(x)$ and $y = g(x)$ is given by $A = \int_a^b [f(x) - g(x)] \, dx$.
• Integration of $\frac{1}{x}$: $\int \frac{1}{x} \, dx = \ln|x| + C$.
• Power Rule for Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

Step-by-Step Solution

Step 1: Identify the Upper and Lower Curves and Limits of Integration

We are given the region defined by $\frac{a}{x^2} \leq y \leq \frac{1}{x}$, $1 \leq x \leq 2$, and $0 < a < 1$. This tells us:

• The upper curve is $f(x) = \frac{1}{x}$.
• The lower curve is $g(x) = \frac{a}{x^2}$.
• The limits of integration are $x = 1$ and $x = 2$.

We also need to verify that $f(x) \ge g(x)$ on the interval $[1, 2]$. This means we need to check if $\frac{1}{x} \ge \frac{a}{x^2}$ for $1 \le x \le 2$. Multiplying both sides by $x^2$ (since $x^2 > 0$) gives $x \ge a$. Since $1 \le x \le 2$ and $0 < a < 1$, the inequality $x \ge a$ holds true.

Step 2: Set Up the Definite Integral

The area of the region is given by the integral of the difference between the upper and lower curves, from the lower limit to the upper limit: $A = \int_1^2 \left( \frac{1}{x} - \frac{a}{x^2} \right) \, dx$

Step 3: Evaluate the Integral

We integrate the expression term by term: $A = \int_1^2 \left( \frac{1}{x} - a x^{-2} \right) \, dx = \left[ \ln|x| - a \frac{x^{-1}}{-1} \right]_1^2 = \left[ \ln|x| + \frac{a}{x} \right]_1^2$ Now, we evaluate the antiderivative at the upper and lower limits: $A = \left( \ln 2 + \frac{a}{2} \right) - \left( \ln 1 + \frac{a}{1} \right) = \ln 2 + \frac{a}{2} - 0 - a = \ln 2 - \frac{a}{2}$

Step 4: Equate to Given Area and Solve for 'a'

We are given that the area is $\ln 2 - \frac{1}{7}$. Therefore: $\ln 2 - \frac{a}{2} = \ln 2 - \frac{1}{7}$ Subtracting $\ln 2$ from both sides gives: $-\frac{a}{2} = -\frac{1}{7}$ Multiplying both sides by $-2$ gives: $a = \frac{2}{7}$

Step 5: Calculate $7a - 3$

We are asked to find the value of $7a - 3$. Substituting $a = \frac{2}{7}$ into this expression gives: $7a - 3 = 7 \left( \frac{2}{7} \right) - 3 = 2 - 3 = -1$

Common Mistakes & Tips

• Sign Errors: Be careful with signs when evaluating the integral at the limits and when simplifying expressions.
• Incorrect Integration: Double-check the integration formulas, especially for terms like $x^{-2}$.
• Order of Curves: Always verify that the chosen upper curve is indeed greater than or equal to the lower curve on the given interval.

Summary

We found the area between the curves by integrating the difference of the functions over the given interval. We then equated this result to the given area to solve for the parameter 'a'. Finally, we substituted the value of 'a' into the expression $7a - 3$ to obtain the final answer.

The final answer is $\boxed{-1}$, which corresponds to option (D).