1. **State the problem:** We want to find the right Riemann sum for the function $f(x) = 3x + 1$ on the interval $[2, 5]$ using $n$ subintervals, and then compute the limit as $n \to \infty$ to find the exact area under the curve. 2. **Formula for right Riemann sum:** The right Riemann sum is given by $$ R_n = \sum_{i=1}^n f(x_i) \Delta x $$ where $\Delta x = \frac{b - a}{n}$ and $x_i = a + i \Delta x$ are the right endpoints of each subinterval. 3. **Calculate $\Delta x$ and $x_i$:** $$ \Delta x = \frac{5 - 2}{n} = \frac{3}{n} $$ $$ x_i = 2 + i \cdot \frac{3}{n} $$ 4. **Evaluate $f(x_i)$:** $$ f(x_i) = 3\left(2 + \frac{3i}{n}\right) + 1 = 6 + \frac{9i}{n} + 1 = 7 + \frac{9i}{n} $$ 5. **Write the right Riemann sum:** $$ R_n = \sum_{i=1}^n \left(7 + \frac{9i}{n}\right) \cdot \frac{3}{n} = \frac{3}{n} \sum_{i=1}^n \left(7 + \frac{9i}{n}\right) $$ 6. **Separate the sum:** $$ R_n = \frac{3}{n} \left( \sum_{i=1}^n 7 + \sum_{i=1}^n \frac{9i}{n} \right) = \frac{3}{n} \left(7n + \frac{9}{n} \sum_{i=1}^n i \right) $$ 7. **Use formula for sum of first $n$ integers:** $$ \sum_{i=1}^n i = \frac{n(n+1)}{2} $$ 8. **Substitute and simplify:** $$ R_n = \frac{3}{n} \left(7n + \frac{9}{n} \cdot \frac{n(n+1)}{2} \right) = \frac{3}{n} \left(7n + \frac{9(n+1)}{2} \right) = \frac{3}{n} \left(7n + \frac{9n}{2} + \frac{9}{2} \right) $$ $$ = \frac{3}{n} \left(7n + 4.5n + 4.5 \right) = \frac{3}{n} \left(11.5n + 4.5 \right) = 3 \left(11.5 + \frac{4.5}{n} \right) = 34.5 + \frac{13.5}{n} $$ 9. **Compute the limit as $n \to \infty$:** $$ \lim_{n \to \infty} R_n = \lim_{n \to \infty} \left(34.5 + \frac{13.5}{n} \right) = 34.5 $$ **Final answer:** The area under $f(x) = 3x + 1$ on $[2,5]$ is $\boxed{34.5}$.