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The left Riemann sum is a technique used to approximate the integral of a function over a specified interval. The process involves dividing the interval into equal parts called sub-intervals and using the left endpoint of each sub-interval to calculate the area of a series of rectangles. The width of each rectangle is the length of the sub-interval, while the height is the value of the function at the left endpoint. This method is useful when you need a quick approximation of the area under a curve.

In this exercise, the interval \([1, 2]\) is divided into 10 equal parts, each of width \(0.1\). For each sub-interval, the function \(f(t) = 4 - t^2\) is evaluated at the left endpoint. By calculating and summing these areas, you can find the left Riemann sum, denoted as \(L_{10}\). This provides an approximation that tends to underestimate the true value of the integral, especially when the function is decreasing over the interval.

When applied to \(f(t) = 4 - t^2\), each term in the sum is calculated using \(f(1 + i \Delta t) \cdot \Delta t\) for \(i = 0, 1, ..., 9\). By summing these terms, the left Riemann sum gives you a step-wise approximation of the integral based on the values at the left endpoints.

The right Riemann sum is another method for approximating the integral of a function. Unlike the left Riemann sum, this method uses the right endpoint of each sub-interval to compute the height of the rectangles. This slight change can often provide a more accurate approximation depending on the function's behavior within the interval.

In the same interval \([1, 2]\) split into 10 sub-intervals, the right Riemann sum takes the value of the function \(f(t) = 4 - t^2\) at the right endpoint. It calculates the areas of the rectangles as \(f(1 + i \Delta t) \cdot \Delta t\) for \(i = 1, 2, ..., 10\). By adding these areas together, you calculate the right Riemann sum, \(R_{10}\).

When dealing with the function in question, the right Riemann sum may overestimate or provide a close estimate to the actual integral value as it accounts for the decreasing nature of the function more effectively than the left method. This potential for overestimating is generally due to the fact that, unlike its left counterpart, it uses the slightly higher value at the right endpoint.

Integral approximation is crucial for estimating the area under a curve when the exact integral is difficult to compute or when an analytical solution is not feasible. Riemann sums, be it left or right, are among the simplest and most useful strategies to approximate integrals. Whether a left or right approach is used, both methods provide visual and numerical approximations of the actual integral.

Combining the left and right Riemann sums by taking their average results in a more balanced approximation of the integral. This average usually offers a more accurate estimate than either sum alone, as it mitigates the errors from underestimation and overestimation inherent in left and right sums, respectively.

For this problem, the average \(\frac{L_{10} + R_{10}}{2}\) is compared against the given integral value \(1.\overline{66}\). The comparison helps in determining how accurate the estimated value is, based on matching decimal places. This practice is not only important for validating numerical methods but also helps in understanding the efficiency and reliability of Riemann sums as integral approximators.