Problem 3 - Thomas Calculus

Question

In Exercises \(1-4,\) use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. $$f(x)=1 /x \quad \text {between} \quad x=1 \quad \text {and} \quad x=5$$

Step-by-Step Solution

Answer

Lower sums: \(\frac{8}{3}\) and \(\frac{77}{12}\); Upper sums: \(\frac{28}{15}\) and \(\frac{77}{60}\).

Step 1: Define the Interval and Calculate Widths

The function given is \( f(x) = \frac{1}{x} \) and the interval is between \( x = 1 \) and \( x = 5 \). The total interval length is \( 5 - 1 = 4 \). For two rectangles, the width \( \Delta x = \frac{4}{2} = 2 \). For four rectangles, \( \Delta x = \frac{4}{4} = 1 \).

Step 2: Calculate Lower Sum with Two Rectangles

For two rectangles using the lower sum approach, check the function's values at \( x \) points. The two rectangle bases will be defined at \( x = 1 \) and \( x = 3 \). Calculating the values: \( f(1) = 1 \) and \( f(3) = \frac{1}{3} \). Therefore, the lower sum is \( A_{lower2} = 2\times(1) + 2\times\left(\frac{1}{3}\right) = 2 + \frac{2}{3} = \frac{8}{3} \).

Step 3: Calculate Lower Sum with Four Rectangles

For four rectangles using the lower sum, the bases will be set at \( x = 1, 2, 3, \) and \( 4 \). Values: \( f(1) = 1, f(2) = \frac{1}{2}, f(3) = \frac{1}{3}, f(4) = \frac{1}{4} \). So, \( A_{lower4} = 1\times(1) + 1\times\left(\frac{1}{2}\right) + 1\times\left(\frac{1}{3}\right) + 1\times\left(\frac{1}{4}\right) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{77}{12} \).

Step 4: Calculate Upper Sum with Two Rectangles

For two rectangles using the upper sum approach, evaluate the function at the points \( x = 3 \) and \( x = 5 \). Values are: \( f(3) = \frac{1}{3}, f(5) = \frac{1}{5} \). Therefore, \( A_{upper2} = 2\times\left(\frac{1}{3}\right) + 2\times\left(\frac{1}{5}\right) = \frac{2}{3} + \frac{2}{5} = \frac{28}{15} \).

Step 5: Calculate Upper Sum with Four Rectangles

For four rectangles using the upper sum, the points to use are \( x = 2, 3, 4, \) and \( 5 \). Values: \( f(2) = \frac{1}{2}, f(3) = \frac{1}{3}, f(4) = \frac{1}{4}, f(5) = \frac{1}{5} \). Then, \( A_{upper4} = 1\times\left(\frac{1}{2}\right) + 1\times\left(\frac{1}{3}\right) + 1\times\left(\frac{1}{4}\right) + 1\times\left(\frac{1}{5}\right) = \frac{77}{60} \).

Key Concepts

Upper Sum

An upper sum is a method to approximate the area under a curve by using rectangles. In the upper sum, rectangles are chosen so that the top edge is the highest point of the curve over each interval.
This guarantees that each rectangle overshoots the actual curve, resulting in an overestimate of the area.
To understand this, let's look at the steps of calculating the upper sum:

• Divide the interval into equal parts, known as subintervals.
• For each subinterval, use the right endpoint to determine the height of the rectangle, since for a decreasing function like \( f(x) = \frac{1}{x} \), the right endpoint provides the largest value on that interval.
• Calculate the area of each rectangle as "height \( \times \) width" where height is \( f(x) \) at the chosen point.
• Sum the areas of all rectangles to find the total upper sum.

For example, with four rectangles to estimate \( \int_{1}^{5} \frac{1}{x} \ dx \), using the points \( x = 2, 3, 4, \) and \( 5 \), the upper sum is \( A_{upper4} = \frac{77}{60} \).

Lower Sum

The lower sum is another method of estimating the area under a curve, but in contrast to the upper sum, it uses rectangles that underestimate the area. The height of each rectangle is determined by the smallest value of the function over each subinterval, often taken at the left endpoint.

Here's how a lower sum is calculated:

• Subdivide the interval into smaller segments, much like in the upper sum.
• For each segment, choose the minimum function value as the height of that rectangle. For \( f(x) = \frac{1}{x} \) on \( x = 1 \) to \( x = 5 \), use \( x = 1, 2, \) etc., for left endpoints.
• Find the rectangle areas by multiplying the chosen height by the width of the segment.
• Add up all rectangles to get the lower sum.

Using four rectangles and values at \( x = 1, 2, 3, \) and \( 4 \) yields a lower sum of \( A_{lower4} = \frac{77}{12} \).

Finite Approximations

Finite approximations are techniques used to estimate values, such as areas, using a finite number of steps.

For integrals, we often can't compute the exact area under complex curves directly, so we approximate using techniques like Riemann sums.
An approximation:

• Involves dividing an interval into subintervals and using simple geometrical shapes like rectangles.
• As rectangles increase in number, the approximation becomes more accurate.
• Can be refined further by reducing the width of each rectangle, increasing precision.

In this exercise, finite approximations using either lower or upper sums give different estimates for the integral \( \int_{1}^{5} \frac{1}{x} \ dx \). Both methods aim to approach the true value as more rectangles are used.

Definite Integral

The definite integral of a function provides the exact area under the curve from one point to another on the x-axis.

It is essentially the limit of Riemann sums as the number of rectangles increases indefinitely.
Definite integrals are useful because:

• They provide precise calculations of areas bounded by curves.
• Represent accumulated quantities, such as distance, area, and volume, when x changes.
• In this context, \( \int_{1}^{5} \frac{1}{x} \ dx \) would give the exact area between the curve \( f(x) = \frac{1}{x} \) and the x-axis from \( x=1 \) to \( x=5 \).

Definite integrals remove the need for approximations once the limit of sums (infinite rectangles) is calculated.