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The Left Riemann Sum is a method used in numerical integration to approximate the area under a curve. It involves summing up the areas of rectangles formed by using the left endpoints of subintervals within a given range.

For a function like \(s(t)=\sin(t)\) over the interval \([0, 4]\), we divide the interval into 8 equal parts. The width of each subinterval \( \Delta t \) is determined by dividing the total interval length by the number of rectangles, giving \( \Delta t = \frac{4}{8} = 0.5\). This means each rectangle spans a width of 0.5 along the x-axis.

**Calculating Left Riemann Sum:**

• To approximate the area, calculate the value of \(s(t)\) at the left endpoint of each subinterval.
• For example, the points for the left endpoints here are \(t = 0, 0.5, 1, 1.5, 2, 2.5, 3,\) and \(3.5\).
• Calculate \( \sin(t)\) at these points, add them together, and then multiply by \( \Delta t = 0.5\).

This gives an estimated area under the curve of approximately 1.468. The Left Riemann Sum can tend to underestimate the actual area if the function increases over the interval, as rectangles will sit below the curve.

The Right Riemann Sum provides another approach to estimating the area under a curve using rectangles. This method uses the function's values at the right endpoints of each subinterval. It is especially useful when the function decreases over the interval, as it might offer a closer approximation than the left sum.

In the same interval \([0, 4]\), divided into 8 parts, where each subinterval has a width \( \Delta t = 0.5\), the right endpoints will be \(t = 0.5, 1, 1.5, 2, 2.5, 3, 3.5,\) and \(4\).

**Calculating Right Riemann Sum:**

• The area estimation involves calculating \(s(t)\) at each of the right endpoints.
• Sum these values, and multiply by the width \( \Delta t = 0.5\).
• This results in an estimated area of approximately 1.557.

By evaluating the function at right endpoints, the Right Riemann Sum can yield a more accurate estimate when the function is decreasing. In this exercise, it was particularly effective in providing the closest approximation to the actual area.

The Midpoint Riemann Sum offers an alternative and often more accurate method to approximate the area under a curve by evaluating the function at the midpoint of each subinterval. This approach can balance overestimation and underestimation seen in the left and right sums.

For \(s(t)=\sin(t)\) from \(0\) to \(4\), with 8 subdivisions, we continue with \( \Delta t = 0.5\). The midpoints fall at \(t = 0.25, 0.75, 1.25, 1.75, 2.25, 2.75, 3.25,\) and \(3.75\).

**Calculating Midpoint Riemann Sum:**

• Evaluate the function \(s(t)\) at these midpoint positions.
• Add the calculated sine values, then multiply them by the subinterval width \( \Delta t = 0.5\).
• This provides an estimated area that is approximately 1.539.

The Midpoint Riemann Sum is particularly favored for providing more accuracy without biasing towards either overestimation or underestimation, especially effective in intervals where the function has both increasing and decreasing aspects.