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The left Riemann sum is a method to approximate the value of an integral by using the function values at the left endpoints of subintervals. This method is particularly useful when you need a quick estimation of the integral, especially if you don't have access to the tools needed for more precise calculations.

For the given problem, we divide the interval \( [2, 3] \) into \( n = 2 \) subintervals. The width of each subinterval is: \[ \text{width} = \frac{3-2}{2} = 0.5 \]

Since we use the left endpoints for our calculations: \[ x_0 = 2, \ x_1 = 2 + 0.5 = 2.5 \]

The left Riemann sum is then: \[ \text{sum} = f(x_0) \times \text{width} + f(x_1) \times \text{width} = e^2 \times 0.5 + e^{2.5} \times 0.5 \]

This approach will give an underestimate of the actual integral because the function \( y = e^x \) is increasing throughout the interval. The rectangles formed will always lie below the curve.

The right Riemann sum offers another method for approximating the value of an integral, differing from the left Riemann sum by using the function values at the right endpoints of subintervals. In situations involving increasing functions, this method typically provides an overestimate of the integral.

For the same interval \( [2, 3] \) divided into \( n = 2 \) subintervals, the right endpoints are: \[ x_1 = 2.5, \ x_2 = 2.5 + 0.5 = 3 \]

The right Riemann sum is calculated as: \[ \text{sum} = f(x_1) \times \text{width} + f(x_2) \times \text{width} = e^{2.5} \times 0.5 + e^{3} \times 0.5 \]

This method generally results in an overestimate of the integral because the rectangles will extend above the curve for an increasing function like \( y = e^x \).

The methods of left Riemann sum and right Riemann sum fall under the broader category of integration approximation. These techniques provide valuable ways to estimate the value of definite integrals, and are especially helpful when dealing with complicated functions or when exact integration is difficult.

There are several key points to remember about these methods:

• Both left and right Riemann sums rely on dividing the interval of integration into smaller subintervals (rectangles).
• The width of each subinterval is calculated as: \[ \text{width} = \frac{b-a}{n}, \ \text{where } a \text{ and } b \text{ are the limits of integration and } n \text{ is the number of subintervals.} \]
• For the left Riemann sum, the function value at the left endpoint of each subinterval is used.
  For the right Riemann sum, the function value at the right endpoint of each subinterval is used.
• When dealing with an increasing function like \( y = e^x \), the left Riemann sum provides an underestimate, while the right Riemann sum provides an overestimate of the actual integral.

These methods are foundational in calculus and serve as stepping stones to more accurate techniques such as Simpson's Rule and the Trapezoidal Rule. Whether you are estimating an integral by hand or using software, understanding these basic concepts is essential for more advanced studies.