Problem 1 - Thomas Calculus

Question

In Exercises \(1-4,\) use finite approximations to estimate the area under the graph of the function using $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ \(f(x)=x^{2}\) between \(x=0\) and \(x=1\)

Step-by-Step Solution

Answer

Lower sums: 0.125 (2 rectangles), 0.21875 (4 rectangles); Upper sums: 0.625 (2 rectangles), 0.46875 (4 rectangles).

Step 1: Determine the interval width for two rectangles

Since we're using two rectangles of equal width across the interval from \( x = 0 \) to \( x = 1 \), we divide the interval into \( n = 2 \) parts. The width \( \Delta x \) for each rectangle is computed as \( \Delta x = \frac{1-0}{2} = 0.5 \).

Step 2: Lower sum with two rectangles

In a lower sum, we take the minimum value of the function in each subinterval. For \( x^2 \) in \([0, 0.5)\) and \([0.5, 1)\), the minimum values are \( f(0) = 0^2 = 0 \) and \( f(0.5) = 0.5^2 = 0.25 \), respectively. Therefore, the lower sum with two rectangles is:\[\text{Lower Sum with 2 Rectangles} = (0)(0.5) + (0.25)(0.5) = 0 + 0.125 = 0.125\]

Step 3: Lower sum with four rectangles

Now divide the interval into 4 equal parts. The width of each rectangle is \( \Delta x = \frac{1}{4} = 0.25 \). The intervals are \([0, 0.25), [0.25, 0.5), [0.5, 0.75), [0.75, 1)\), and the minimum values in these intervals are \( f(0)^2 = 0 \), \( f(0.25)^2 = 0.0625 \), \( f(0.5)^2 = 0.25 \), and \( f(0.75)^2 = 0.5625 \). Sum these:\[\text{Lower Sum with 4 Rectangles} = 0(0.25) + 0.0625(0.25) + 0.25(0.25) + 0.5625(0.25) = 0 + 0.015625 + 0.0625 + 0.140625 = 0.21875\]

Step 4: Upper sum with two rectangles

For the upper sum with 2 rectangles, we use the maximum value of the function in each interval. The maximum values are at \( f(0.5)^2 = 0.25 \) and \( f(1)^2 = 1 \). Therefore:\[\text{Upper Sum with 2 Rectangles} = 0.25(0.5) + 1(0.5) = 0.125 + 0.5 = 0.625\]

Step 5: Upper sum with four rectangles

Divide into 4 rectangles with width \( \Delta x = 0.25 \) again. The maximum values are \( f(0.25)^2 = 0.0625 \), \( f(0.5)^2 = 0.25 \), \( f(0.75)^2 = 0.5625 \), and \( f(1)^2 = 1 \). Thus, the upper sum is:\[\text{Upper Sum with 4 Rectangles} = 0.0625(0.25) + 0.25(0.25) + 0.5625(0.25) + 1(0.25) = 0.015625 + 0.0625 + 0.140625 + 0.25 = 0.46875\]

Key Concepts

Finite Approximation

Finite approximation is a method used to estimate the area under a curve on a graph. This is particularly useful when dealing with complex functions or functions without a definite integral. The idea is to approximate the area using a finite number of shapes, such as rectangles, which are easy to calculate. By dividing the total area into smaller sections, we can get a better estimation. The more rectangles we use, the closer our approximation is to the actual area. In our exercise, we use two main concepts of approximation: the lower sum and the upper sum. Finite approximation is fundamental in calculus, laying down the groundwork for integral calculus, where the goal is to find exact areas. It helps students visualize how integrals work using finite, understandable steps.

Lower Sum

The lower sum is a method of approximation where we construct rectangles using the minimum value of the function within each subinterval. This means that each rectangle fits under the curve, giving us an underestimate of the area. For instance, when dividing the interval from 0 to 1 into subintervals of equal width, we calculate the area of each rectangle by identifying the lowest value of the function in each subinterval.

• When using two rectangles, the width of each is 0.5.
• The minimum values for the intervals [0, 0.5) and [0.5, 1) are 0 and 0.25, respectively.

The lower sum provides a conservative estimate by calculating the sum of areas of rectangles placed entirely below the curve. Increasing the number of rectangles leads to a more accurate approximation.

Upper Sum

The upper sum is another important approximation technique, but it looks at the maximum value within each subinterval to form the rectangles. This approach leads to an overestimate since the rectangles often extend above the curve. Using the same example as before:

• The maximum values for the segments, when divided into two intervals, appear at 0.25 and 1.
• For four rectangles, the heights correspond to maximum values in incremental steps, better tailoring the rectangles to the curve’s shape.

While the upper sum gives us a more generous estimate, combining both lower and upper sum calculations can provide a range within which the actual area lies. This insight helps in understanding the behavior of the function over the interval.

Rectangles of Equal Width

Using rectangles of equal width is a critical aspect of finite approximation. This ensures each subinterval is evenly spaced, simplifying calculations and comparisons of areas.The width of these rectangles, designated as \( \Delta x \), is determined by dividing the total range of the interval by the number of rectangles you choose to use.

• For example, if we choose two rectangles over the interval from 0 to 1, each has a width of 0.5.
• Choosing four rectangles would mean each has a width of 0.25.

Equal widths not only simplify the process, but they also create uniformity in the approximation method, making the summation straightforward. This consistent partitioning is essential for systematically estimating areas under more complex curves in calculus.