To solve this problem, let's follow the steps to construct the Riemann sum and take its limit as $n \to \infty$.

Step 1: Define the interval and subintervals

The function given is: $f(x) = 2x$ on the interval $[2, 6]$, with $a = 2$ and $b = 6$.

To divide the interval $[2, 6]$ into $n$ equal subintervals, each subinterval will have width: $\Delta x = \frac{b - a}{n} = \frac{6 - 2}{n} = \frac{4}{n}.$

Step 2: Define the Right Endpoint of Each Subinterval

Using the right endpoint for each subinterval $c_k$, we have: $c_k = a + k \cdot \Delta x = 2 + k \cdot \frac{4}{n}.$

Step 3: Set Up the Riemann Sum

The Riemann sum $R_n$ with $n$ subintervals and using the right endpoints is: $R_n = \sum_{k=1}^n f(c_k) \Delta x.$ Substituting $f(x) = 2x$, $f(c_k) = 2 \cdot c_k$: $R_n = \sum_{k=1}^n 2 \left(2 + \frac{4k}{n}\right) \cdot \frac{4}{n}.$

Expanding this, we get: $R_n = \sum_{k=1}^n \left(4 + \frac{8k}{n}\right) \cdot \frac{4}{n}.$ $R_n = \sum_{k=1}^n \left(\frac{16}{n} + \frac{32k}{n^2}\right).$ $R_n = \sum_{k=1}^n \frac{16}{n} + \sum_{k=1}^n \frac{32k}{n^2}.$

Step 4: Simplify the Sum

Now, we calculate each sum separately:

1. First Sum: $\sum_{k=1}^n \frac{16}{n} = 16$, since this is a constant term multiplied $n$ times and divided by $n$.
2. Second Sum: For $\sum_{k=1}^n \frac{32k}{n^2}$, we use the formula for the sum of the first $n$ natural numbers: $\sum_{k=1}^n k = \frac{n(n+1)}{2}$: $\sum_{k=1}^n \frac{32k}{n^2} = \frac{32}{n^2} \cdot \frac{n(n+1)}{2} = \frac{32(n+1)}{2n} = \frac{16(n+1)}{n}.$

Thus: $R_n = 16 + \frac{16(n+1)}{n} = 16 + 16 + \frac{16}{n} = 32 + \frac{16}{n}.$

Step 5: Take the Limit as $n \to \infty$

To find the area under the curve, we take the limit of $R_n$ as $n \to \infty$: $\lim_{n \to \infty} R_n = \lim_{n \to \infty} \left(32 + \frac{16}{n}\right) = 32.$

Conclusion

The area under the curve $f(x) = 2x$ over the interval $[2, 6]$ is: $\boxed{32}.$

Would you like more details on any of these steps or have further questions?

Here are five related questions for further practice:

1. How would the Riemann sum change if we used the left endpoint instead of the right endpoint?
2. What would the Riemann sum look like for a function $f(x) = x^2$ over the interval $[1, 5]$?
3. Can you derive the Riemann sum formula for $f(x) = 3x + 1$ over an interval $[0, 4]$ using $n$ subintervals?
4. How would you compute the area under a curve if the interval were split into unequal subintervals?
5. What happens if the function is discontinuous within the interval—how would this affect the Riemann sum?

Tip: Practicing with different types of functions, intervals, and endpoints can help deepen your understanding of Riemann sums and integral approximations.