Question

Rectangular Approximation Methods Show that if \(f\) is a nonnegative function on the interval \([a, b]\) and the line \(x=(a+b) / 2\) is a line of symmetry of the graph of \(y=f(x)\) then \(L R A M_{n} f=\operatorname{RRAM}_{n} f\) for every positive integer \(n .\)

Short answer

For the given symmetric function, the left and right rectangular approximations are equivalent for any number of subdivisions. Hence, \(LRAM_{n} f=\operatorname{RRAM}_{n} f\) for every positive integer \(n\).

Step-by-step solution

Principle of Rectangular Approximation Methods (RAM)

The Rectangular Approximation Method (RAM) is a technique for approximating the total area under a curve or the integral of a function. LRAM and RRAM, respectively, use the leftmost and rightmost points of each subinterval.

Consider n subdivisions of [a, b]

Assume without loss of generality that \([a, b]\) is divided into n equal subintervals. Because \(f\) is symmetric about \(x = (a+b) / 2\), for each subinterval, the value of \(f\) at the left endpoint is equal to the value of \(f\) at the corresponding right endpoint.

Show LRAM = RRAM

For LRAM, the value of the function at the leftmost point of the subinterval is taken, while in RRAM, the value of the function at the rightmost point of the subinterval is taken. Because \(f\) is symmetric and the subintervals are equally spaced, for every subinterval \(I_i\) starting from the left, there exists a corresponding subinterval \(I_j\) from the right where the height of the rectangle in the LRAM will be equal to the height of the rectangle in the RRAM. So, the sum of the areas of the rectangles in the LRAM, which gives \(LRAM_n f\), equals the sum of the areas of the rectangles in the RRAM, which gives \(RRAM_n f\). Therefore, \(LRAM_n f = RRAM_n f\), for every positive integer n.

Key concepts

LRAM (Left Rectangular Approximation Method)

Often in calculus, we need to estimate the area under a curve before we learn the definitive techniques of integrating functions. LRAM stands for Left Rectangular Approximation Method and is a tool for estimating the value of a definite integral.

Consider a curve mapped by a function, and you are interested in the area beneath this curve over a specific interval, say from point 'a' to point 'b'. In LRAM, this interval is divided into 'n' equal subintervals. For each division, a rectangle is constructed with its height equal to the value of the function at the left endpoint of the subinterval. The width of each rectangle is the length of the subinterval. After constructing rectangles on all subintervals, the sum of their areas offers an approximation to the true area under the curve.

One unique feature of LRAM is its potential overestimation or underestimation of the total area. The approximation depends highly on the behavior of the function on the specified interval—if the function is increasing, LRAM tends to underestimate, while it overestimates for a decreasing function.

RRAM (Right Rectangular Approximation Method)

The counterpart to LRAM is the Right Rectangular Approximation Method, or RRAM. As the name suggests, RRAM is almost a mirror image of LRAM in terms of approach but instead of considering the left endpoints to determine the height of the rectangles, it considers the right endpoints.

When the interval from 'a' to 'b' is divided into 'n' subintervals, rectangles are constructed such that the height of each rectangle corresponds to the value of the function at the right endpoint of the respective subinterval. As with LRAM, the approximation's accuracy will depend on whether the function is increasing or decreasing. RRAM tends to overestimate the area under an increasing curve and underestimate it when the curve is decreasing. This method is particularly useful when you have a function that's easy to evaluate at the right endpoints of intervals.

Integral Approximation

Integral Approximation encompasses a variety of methods, including LRAM and RRAM, for estimating the value of integrals, which in geometrical terms translates to calculating the area under curves. Other methods like the midpoint rule or trapezoidal rule also fall under this category.

While LRAM and RRAM use rectangles to model the areas, other techniques may use shapes like trapezoids or consider the midpoint of subintervals for a more accurate approximation. Integral approximation is especially valuable when dealing with functions that are difficult to integrate analytically, or when a quick estimate is required. Advances in numerical methods and computational power have enabled us to approximate definite integrals with great precision, but understanding the underlying principles remains essential for interpreting and validating these results.

Function Symmetry

Symmetry in functions is a foundational concept that has significant implications in the study of calculus, particularly in simplifying the process of integration and approximation of areas.

If a function is symmetric about a vertical line, like the y-axis or the line described by the equation x = (a+b)/2, it means that for every point on the function to the left of the line of symmetry, there is a mirror image point on the right. This symmetry leads to an interesting property particularly relevant to Rectangular Approximation Methods: for a symmetric function over a symmetric interval, LRAM and RRAM will yield the same result. This is because the heights of the rectangles on one side of the line of symmetry will exactly match their counterparts on the other side, even if the rectangles are on opposite ends of each subinterval. Understanding this concept can not only save time in calculations but also deepen insights into the geometric properties of functions and their graphs.