Question

Rectangular Approximation Methods Prove or disprove the following statement: MRAM \(_{n}\) is always the average of LRAM \(_{n}\) and \(\operatorname{RRAM}_{n}\).

Short answer

The statement 'MRAM \(_{n}\) is always the average of LRAM \(_{n}\) and \(\operatorname{RRAM}_{n}\)' is disproved. The Midpoint Rectangular Approximation Method (MRAM) is not the average of Left and Right Rectangular Approximation Methods (LRAM and RRAM).

Step-by-step solution

Define LRAM, RRAM, and MRAM

LRAM uses the left end point of each interval as the height of the rectangle, RRAM uses the right end point, and MRAM uses the midpoint. If we define \(f(x)\) as the function we are evaluating and Δx as the width of the intervals, then the equations for each method would look as follows: \[LRAM_n = \Delta x [f(a) + f(a + \Delta x) + \ldots + f(a + (n-1)\Delta x)] \] \[RRAM_n = \Delta x [f(a + \Delta x) + f(a + 2\Delta x) + \ldots + f(a + n\Delta x)]\] \[MRAM_n = \Delta x [f(a + \frac{1}{2}\Delta x) + f(a + \frac{3}{2}\Delta x) + \ldots + f(a + (n-1+\frac{1}{2})\Delta x)] \]

Show that MRAM_n is the average of LRAM_n and RRAM_n

If we add the equations of LRAM and RRAM and divide by 2, that gives us average approximation method. Here: \[ \frac{LRAM_n + RRAM_n}{2} = \Delta x [f(a)+\frac{1}{2}f(a+\Delta x) + \frac{1}{2}f(a+2\Delta x) +\ldots+ \frac{1}{2}f(a + n\Delta x)] \] Thus, indeed Average(LRAM, RRAM) is not equal to MRAM. Hence, the statement MRAM \(_{n}\) is always the average of LRAM \(_{n}\) and \(\operatorname{RRAM}_{n}\) is not true.

Key concepts

LRAM (Left Rectangular Approximation Method)

The Left Rectangular Approximation Method (LRAM) is a technique used to estimate the area under a curve on a graph. This method assumes that the value of a function at the left endpoint of an interval represents the height of the rectangle on that interval. Imagine you are slicing the area under the curve into vertical strips and then using the left edge of each strip to create a rectangle.

For a function f(x) and intervals of equal width Δx, the LRAM can be formulated as:
\[LRAM_n = \Delta x [f(a) + f(a + \Delta x) + \ldots + f(a + (n-1)\Delta x)] \]
Here, a is the starting point of the interval, and n represents the number of intervals. This method can underestimate the area if the function is increasing and overestimate if the function is decreasing over the interval.

RRAM (Right Rectangular Approximation Method)

Conversely, the Right Rectangular Approximation Method (RRAM) mirrors LRAM but focuses on the right endpoint of each interval to define the rectangle's height. This means if you have the same vertical strips under the curve as with LRAM, RRAM uses the right edge to form rectangles.

Mathematically expressed, the formula for RRAM is:
\[RRAM_n = \Delta x [f(a + \Delta x) + f(a + 2\Delta x) + \ldots + f(a + n\Delta x)] \]
In this case, the approximation can overestimate the area under a curve if it's increasing and underestimate if it's decreasing over each interval. These characteristics are essential to remember when comparing the accuracy of rectangular approximation methods.

MRAM (Midpoint Rectangular Approximation Method)

The Midpoint Rectangular Approximation Method (MRAM) offers a different approach by taking the midpoint of each interval as the height of the associated rectangle. This generally provides a more balanced estimate than LRAM or RRAM alone because it averages the errors made by the other two methods across intervals where the function is either increasing or decreasing.

The MRAM is calculated with the formula:
\[MRAM_n = \Delta x [f(a + \frac{1}{2}\Delta x) + f(a + \frac{3}{2}\Delta x) + \ldots + f(a + (n-1+\frac{1}{2})\Delta x)] \]
While MRAM is often a better approximation than either LRAM or RRAM, it is not always simply their mathematical average, as detailed analyses and examples can reveal.

Integral Approximation

Integral approximation is a broader term that includes methods like LRAM, RRAM, and MRAM to estimate the value of definite integrals. The goal here is to calculate the area under a curve between two points, which is a common problem in calculus. Rectangular methods offer a straightforward way to do this by summing up the areas of rectangles that approximate the curve.

However, the true area is defined as the limit of these sums as the number of rectangles increases indefinitely, which leads us to the concept of a definite integral. If we increase the number of intervals n to infinity, thus making Δx approaches zero, these approximations tend to the exact area under the curve, which is the integral of the function:
\[\int_a^b f(x) dx \]
Practically, integral approximation methods are a foundation for numerical integration, helping solve integrals that lack closed-form antiderivatives or when quick, yet accurate, estimates are needed.