To find the left-hand and right-hand Riemann sums for the integral of $y = e^x$ over the interval $[1, 3]$, you need to approximate the area under the curve by dividing the interval into $n$ subintervals and then applying the Riemann sum formulas.

1. Partition the interval: Divide $[1, 3]$ into $n$ subintervals of equal width, where each subinterval width $\Delta x$ is given by: $\Delta x = \frac{3 - 1}{n} = \frac{2}{n}$

2. Left Riemann Sum (Lower Sum): For the left Riemann sum, evaluate $y = e^x$ at the left endpoint of each subinterval. If we let $x_i = 1 + (i-1)\Delta x$, the left Riemann sum $L_n$ is: $L_n = \sum_{i=1}^{n} e^{x_{i-1}} \cdot \Delta x = \sum_{i=1}^{n} e^{1 + (i-1)\frac{2}{n}} \cdot \frac{2}{n}$

3. Right Riemann Sum (Upper Sum): For the right Riemann sum, evaluate $y = e^x$ at the right endpoint of each subinterval. If we let $x_i = 1 + i\Delta x$, the right Riemann sum $R_n$ is: $R_n = \sum_{i=1}^{n} e^{x_i} \cdot \Delta x = \sum_{i=1}^{n} e^{1 + i\frac{2}{n}} \cdot \frac{2}{n}$

4. Taking the Limit: To find the exact value of the integral as $n \to \infty$, compute the integral directly: $\int_{1}^{3} e^x \, dx = \left[ e^x \right]_1^3 = e^3 - e^1$

Would you like further details on these calculations, or do you have any specific questions?

Here are some related questions to deepen your understanding:

1. How does the Riemann sum change as $n$ increases?
2. What is the significance of the difference between the left-hand and right-hand Riemann sums?
3. How would you approximate this integral using a midpoint Riemann sum?
4. What happens if you approximate this integral using trapezoidal sums instead?
5. Why does taking the limit as $n \to \infty$ provide the exact area?

Tip: Increasing the number of subintervals $n$ improves the accuracy of the Riemann sum approximation, bringing it closer to the exact integral value.