To estimate the area under the graph of $f(x) = 2x$ using four approximating rectangles and right endpoints, we proceed as follows:

Step 1: Divide the interval

The interval of interest is from $x = 1$ to $x = 2$. Divide this interval into 4 equal subintervals:

$\text{Width of each subinterval: } \Delta x = \frac{2 - 1}{4} = 0.25.$

The subintervals are: [ [1, 1.25], [1.25, 1.5], [1.5, 1.75], [1.75, 2]. ]

Step 2: Determine the right endpoints

The right endpoints of these subintervals are: $x = 1.25, 1.5, 1.75, 2.$

Step 3: Evaluate $f(x)$ at the right endpoints

$f(1.25) = 2(1.25) = 2.5, \quad f(1.5) = 2(1.5) = 3, \quad f(1.75) = 2(1.75) = 3.5, \quad f(2) = 2(2) = 4.$

Step 4: Calculate the area of each rectangle

The area of each rectangle is given by: $\text{Area of rectangle} = f(x_i) \cdot \Delta x.$ Thus: $\text{Area for } x = 1.25: \, 2.5 \cdot 0.25 = 0.625,$ $\text{Area for } x = 1.5: \, 3 \cdot 0.25 = 0.75,$ $\text{Area for } x = 1.75: \, 3.5 \cdot 0.25 = 0.875,$ $\text{Area for } x = 2: \, 4 \cdot 0.25 = 1.$

Step 5: Add up the areas

The total estimated area is: $\text{Total area} = 0.625 + 0.75 + 0.875 + 1 = 3.25.$

Final Answer:

The estimated area under the graph is: $\boxed{3.2500}$

Would you like details on this process or further explanation? Here are some related questions to explore:

1. How does changing the number of rectangles affect the accuracy of the estimate?
2. What is the result if left endpoints are used instead of right endpoints?
3. How do you compute the exact area under the curve using integration?
4. What happens if the function is non-linear instead of linear?
5. Can we generalize this method for any function $f(x)$ over a specified interval?

Tip: Using more rectangles (i.e., decreasing $\Delta x$) generally improves the accuracy of the estimate.