Question

In Exercises \(9-12,\) use RAM to estimate the area of the region enclosed between the graph of \(f\) and the \(x\) -axis for \(a \leq x \leq b\) . $$f(x)=\frac{1}{x}, \quad a=1, \quad b=3$$

Short answer

The area under the curve \(f(x)=\frac{1}{x}\) from \(x=1\) to \(x=3\) can be approximated using the Right Antiderivative Method (RAM) as \(\sum_{i=1}^{n} (2/n) * 1/(1 + i*(2/n))\). The better the approximation desired, the larger \(n\) should be.

Step-by-step solution

Compute the width of the rectangles

The width of each rectangle can be determined by subtracting the lower limit of integration from the upper limit, and dividing by the desired number of rectangles. For this case, the width would be \((b - a)/n = (3 - 1)/n = 2/n\).

Compute the height of the rectangles

The height of each rectangle is simply the function value at the right endpoint. Here, we write the general formula for the right endpoint: \(x_i = a + i*w\) where \(w\) is the width of each rectangle, \(i\) is the index of the rectangle, and \(a = 1\). Substituting the formula into \(f(x)\), we get \(f(x_i) = 1/(1+i*(2/n))\).

Calculate Area using RAM

The total area under the curve can be approximated by adding together the areas of all \(n\) rectangles. The area of each rectangle is given by their width times their height, i.e., \(Area_i = w*f(x_i)\). Adding up the areas of all rectangles gives us the total area: \(Area = \sum_{i=1}^{n} w*f(x_i) = \sum_{i=1}^{n} (2/n) * 1/(1 + i*(2/n))\). The closer \(n\) is to infinity, the better the approximation (or we can use a large enough \(n\) to get a reasonable approximation).

Key concepts

Integral Approximation

The concept of integral approximation is a cornerstone in understanding calculus and its applications. When you're working with a complex shape under a curve and between the x-axis, such as the one given by the function \(f(x) = \frac{1}{x}\), calculating the exact area can be challenging. To solve this, mathematicians rely on approximation methods.

Instead of finding a precise area, integral approximation techniques allow you to estimate the area by breaking it down into simpler, more manageable shapes. One of the most commonly used techniques is to approximate the area using rectangles, which leads us into the rectangular approximation method (RAM). The finer the division of the rectangles, the more accurate the approximation of the integral, or the actual area under the curve.

Rectangular Approximation Method

The rectangular approximation method or RAM is a practical approach to estimate the area under a curve—the integral—particularly when the function is too complex for an exact integration or when a quick estimation is required. According to RAM, you divide the area into 'n' rectangles, compute their individual areas, and then sum them up to get the total area.

For the function \(f(x) = \frac{1}{x}\), the height of each rectangle is determined by the function's value at specific points, and the width is consistent across all rectangles. The selection of points where the function's value is assessed can be at the left, midpoint, or right of the rectangle's base. In our case, we are using the right endpoints to evaluate the function's value for each rectangle. As

• Step 1

outlines, calculating the width is straightforward and involves simple arithmetic.

• Step 2

and

• Step 3

then focus on determining the height and subsequently the area of each individual rectangle.

Definite Integrals

A definite integral of a function provides the exact area under the curve bounded by the function, the x-axis, and vertical lines at the limits of integration, typically denoted as 'a' and 'b'. It's represented by the integral sign with lower and upper limits, \( \int_a^b f(x) \, dx \). For functions that are integrable in the traditional sense, calculating a definite integral gives you the exact value of the enclosed area.

However, in some cases where the exact calculation is not feasible or when a numerical answer is needed quickly, approximation methods like RAM come into play. RAM can be thought of as a practical means to obtain an estimated value of the definite integral by adding the areas of rectangles, as discussed in the steps of the RAM process.

Function Evaluation

Evaluating a function is a fundamental skill in calculus and is crucial to the rectangular approximation method. It involves finding the value of the function at specific points within the domain of interest. For our function \(f(x) = \frac{1}{x}\), evaluating it requires substituting the x-values into the function to get the respective y-values, or heights of the rectangles.

When approximating the integral with RAM, as shown in

• Step 2

, you evaluate the function at the right-side endpoints of each subinterval, which is represented by \(x_i\). The process may seem daunting at first, but it is essentially about applying the function to a series of x-values dictated by the width of the rectangles and their respective positions along the x-axis, then multiplying each resulting value by the width to find the area contributed by each rectangle.