To estimate the area under the graph of $f(x) = \frac{1}{x}$ from $x = 1$ to $x = 2$ using four approximating rectangles with right endpoints, we follow these steps:

1. Interval Width ($\Delta x$): The interval from $x = 1$ to $x = 2$ is divided into four equal parts, so: $\Delta x = \frac{2 - 1}{4} = 0.25$

2. Right Endpoints: With $\Delta x = 0.25$, the right endpoints are: $x_1 = 1.25, \quad x_2 = 1.5, \quad x_3 = 1.75, \quad x_4 = 2$

3. Function Values at Right Endpoints: $f(1.25) = \frac{1}{1.25} = 0.8$ $f(1.5) = \frac{1}{1.5} \approx 0.6667$ $f(1.75) = \frac{1}{1.75} \approx 0.5714$ $f(2) = \frac{1}{2} = 0.5$

4. Approximate Area: The area is approximated by the sum of the areas of the rectangles, each with width $\Delta x = 0.25$ and height $f(x_i)$: $\text{Area} \approx 0.25 \times (0.8 + 0.6667 + 0.5714 + 0.5) = 0.25 \times 2.5381 = 0.6345$

Your answer of $0.6345$ is indeed correct.

Would you like a breakdown of any part of this solution, or have further questions?

1. What if the number of rectangles is increased for better accuracy?
2. How would the answer change if we used left endpoints instead?
3. How do we find the exact area under the curve analytically?
4. What would be the midpoint approximation for this interval?
5. How does the area approximation compare with the integral's actual value?

Tip: Increasing the number of rectangles generally improves the approximation accuracy by reducing the error.