Problem 2 - Thomas Calculus

Question

In Exercises \(1-4,\) use finite approximations to estimate the area under the graph of the function using $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ \(f(x)=x^{3}\) between \(x=0\) and \(x=1\)

Step-by-Step Solution

Answer

a. 0.0625, b. 0.140625, c. 0.5625, d. 0.390625.

Step 1: Define Interval and Width for Two Rectangles

The interval of interest is from \(x = 0\) to \(x = 1\). For two rectangles, the width of each rectangle is calculated as \( \frac{1-0}{2} = 0.5\).

Step 2: Lower Sum with Two Rectangles

For the lower sum, choose the left endpoints. For \( [0, 0.5] \), the height is \(f(0) = 0^3 = 0\). For \( [0.5, 1] \), the height is \(f(0.5) = (0.5)^3 = 0.125\). The lower sum is \(0 \times 0.5 + 0.125 \times 0.5 = 0.0625\).

Step 3: Lower Sum with Four Rectangles

Each rectangle has a width of \( \frac{1-0}{4} = 0.25\). The left endpoints are at \(0, 0.25, 0.5,\) and \(0.75\). Heights are \(f(0) = 0\), \(f(0.25) = 0.015625\), \(f(0.5) = 0.125\), and \(f(0.75) = 0.421875\). Lower sum is \(0 \times 0.25 + 0.015625 \times 0.25 + 0.125 \times 0.25 + 0.421875 \times 0.25 = 0.140625\).

Step 4: Upper Sum with Two Rectangles

For the upper sum, choose the right endpoints. The right endpoints for \( [0, 0.5] \) and \( [0.5, 1] \) are \(0.5\) and \(1\) respectively. The heights are \(f(0.5) = 0.125\) and \(f(1) = 1\). The upper sum is \(0.125 \times 0.5 + 1 \times 0.5 = 0.5625\).

Step 5: Upper Sum with Four Rectangles

The upper sum considers the right endpoints at \(0.25, 0.5, 0.75, 1\). The heights are \(f(0.25) = 0.015625\), \(f(0.5) = 0.125\), \(f(0.75) = 0.421875\), and \(f(1) = 1\). Thus, the upper sum is \(0.015625 \times 0.25 + 0.125 \times 0.25 + 0.421875 \times 0.25 + 1 \times 0.25 = 0.390625\).

Key Concepts

Finite Approximations

Finite Approximations provide a method to estimate the integral, or the area under a curve, when the exact value is not easily calculable. In the scenario provided with the function \(f(x) = x^3\) from \(x = 0\) to \(x = 1\), finite approximations help visualize and calculate areas using simple shapes, such as rectangles. By dividing this interval into equal sections, we can calculate the sum of the areas of rectangles that closely represent the total area under the curve.

Finite approximations are powerful because they allow complex areas to be broken down into manageable computations. As the number of rectangles increases, the approximation becomes more accurate.

• Using two rectangles is an initial, simpler approximation that provides a basic idea of the area.
• Four rectangles usually lead to a more precise approximation compared to two rectangles, as they better model the curve's inclines and declines.

Mastering this method opens up deeper understanding and application of calculus concepts, which are foundational for more advanced mathematical studies.

Lower Sums

In the context of estimating areas under a curve, lower sums are calculated by taking the lowest point (left endpoint) of each subinterval when forming rectangles. Given the function \(f(x) = x^3\), utilizing lower sums means the height of each rectangle is determined by evaluating the function at the left endpoint of each subinterval. This inherently results in the rectangles underestimating the true area under the curve.

For example, for two rectangles on the interval \([0, 1]\):

• First rectangle: left endpoint at \(x = 0\), height \(f(0) = 0\), width \(0.5\).
• Second rectangle: left endpoint at \(x = 0.5\), height \(f(0.5) = 0.125\), width \(0.5\).

The total lower sum in this case, \(0.0625\), is the sum of the areas of these two rectangles, calculated as height times width. When using four rectangles, more details of the curve are captured, improving the estimation of the area beneath \(f(x)\). Lower sums serve as a conservative estimate, as they tend to be less than or equal to the actual integral.

Upper Sums

Upper sums are determined by selecting the largest point (right endpoint) of each subinterval when evaluating the height of the rectangles. For the function \(f(x) = x^3\) over \([0, 1]\), this means using the value of the function at the right endpoint to determine the rectangle's height, typically leading to an overestimation of the area under the curve.

For an approximation with two rectangles:

• First rectangle: right endpoint at \(x = 0.5\), height \(f(0.5) = 0.125\), width \(0.5\).
• Second rectangle: right endpoint at \(x = 1\), height \(f(1) = 1\), width \(0.5\).

The calculated upper sum of \(0.5625\) is usually larger than the actual area. With four rectangles, the upper sum becomes precise with improved definition as it more effectively captures the shape of the curve. Although providing an inflated estimate, upper sums are useful for bounding the integral from above, giving a maximum area concern.

Area Under Curve

The area under a curve is a fundamental concept in calculus representing the total accumulation of quantities, such as distance, time, or any measure over a given period. Evaluating the area under the curve \(f(x) = x^3\) from \(x=0\) to \(x=1\) involves integrating the function over that interval.

When exact integration is complex or the function is only partially defined, finite approximations such as lower and upper sums provide practical solutions to approximate these areas.

• Lower sums offer an estimate below the true value, ensuring a conservative approximation.
• Upper sums provide an estimation that often surpasses the true area, acting as an upper boundary.

By incrementally increasing the number of rectangles in our finite approximation, the overall estimate of the area becomes more precise and closer to the true integral. Understanding this concept is vital for handling real-life data approximations and lends itself to further study in integral calculus and its applications.