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When we talk about definite integral approximation, we refer to finding an estimated value of the area under a curve over a specific interval. In calculus, the definite integral represents the exact area under a curve between two points, but exact calculations can be complex or, sometimes, impossible. This is where approximation methods like the Riemann Sum come into play. These methods partition the interval into simpler geometric shapes, such as rectangles or trapezoids, and sum their areas to approximate the integral. It may not be perfect, but with a sufficient number of partitions, the approximation can be very close to the actual area.

The Midpoint Rule is a specific type of Riemann Sum that improves the accuracy of our approximation for a definite integral. Rather than using the left or right endpoints of subintervals to determine the height of the rectangles, the Midpoint Rule takes the value of the function at the midpoint of each subinterval. This tends to give a better estimate because it often balances the overestimate and underestimate that occur with left and right endpoint methods, especially when the function is fairly smooth over the interval. To apply the Midpoint Rule, we divide the total interval into a number of equal parts, find the midpoints for these subintervals, calculate the function value at these midpoints, and then sum up all the areas of rectangles formed by these heights and the uniform width \( \Delta x \).

The Rectangular Approximation Method (RAM) refers to a variety of techniques that use rectangles to approximate the area under a curve. This could involve taking the function's value at the left endpoints (Left Riemann Sum), right endpoints (Right Riemann Sum), or the midpoints (Midpoint Rule), of the subintervals as the height of these rectangles. Each rectangle's area is then the product of its height and its width \( \Delta x \) – which is the length of the subinterval. Summing up all these rectangles' areas provides an approximation of the integral. The more subintervals we use, the thinner the rectangles become, and the closer our approximation gets to the true value of the integral.

The integration of the cosine function over an interval can be visualized as the area under the curve of \( y = \cos x\) from one point to another on the x-axis. The cosine function oscillates between -1 and 1, and its integral represents accumulated areas with positive and negative values, corresponding to the parts of the curve above and below the x-axis, respectively. Approximating the integral of the cosine function over an interval with Riemann Sums involves summing the areas of rectangles based on the cosine values at chosen points within the subintervals. For functions like cosine that have regular oscillations, methods like the Midpoint Rule often provide a good estimate for the definite integral, as seen in the example provided with \( f(x) = \cos x \) approximated over \( \left[0, \frac{\pi}{2}\right]\) using midpoints.