Problem 1 - 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. $$ f(x)=x^{2} \text { between } x=0 \text { and } x=1 $$

Step-by-Step Solution

Answer

Lower sums: 0.125 (2 rectangles), 0.21875 (4); Upper sums: 0.625 (2 rectangles), 0.34375 (4).

Step 1: Define the intervals for rectangles

The function is defined on the interval from \(x = 0\) to \(x = 1\). For both parts (a) and (b), we need to partition this interval into equal-width segments. For two rectangles, each rectangle will have a width of \(0.5\), and for four rectangles, the width will be \(0.25\).

Step 2: Calculate lower sum with two rectangles

For the lower sum, we use the left endpoint of each interval. In intervals \([0, 0.5]\) and \([0.5, 1]\), the left endpoints are \(0\) and \(0.5\) respectively. The function values at these points are \(f(0) = 0^2 = 0\) and \(f(0.5) = (0.5)^2 = 0.25\). Thus, the lower sum is:\[ \text{Lower Sum (2 rectangles)} = 0.5 \times 0 + 0.5 \times 0.25 = 0.125 \]

Step 3: Calculate lower sum with four rectangles

For four rectangles, the left endpoints are \(0, 0.25, 0.5,\) and \(0.75\). The function values are \(f(0) = 0, f(0.25) = 0.0625, f(0.5) = 0.25, f(0.75) = 0.5625\). Thus, the lower sum is:\[ \text{Lower Sum (4 rectangles)} = 0.25 \times (0 + 0.0625 + 0.25 + 0.5625) = 0.21875 \]

Step 4: Calculate upper sum with two rectangles

For the upper sum, we use the right endpoint. The right endpoints for intervals \([0, 0.5]\) and \([0.5, 1]\) are \(0.5\) and \(1\) respectively. The function values are \(f(0.5) = 0.25\) and \(f(1) = 1\). Thus, the upper sum is:\[ \text{Upper Sum (2 rectangles)} = 0.5 \times 0.25 + 0.5 \times 1 = 0.625 \]

Step 5: Calculate upper sum with four rectangles

For four rectangles, the right endpoints are \(0.25, 0.5, 0.75,\) and \(1\). The function values are \(f(0.25) = 0.0625, f(0.5) = 0.25, f(0.75) = 0.5625, f(1) = 1\). Thus, the upper sum is:\[ \text{Upper Sum (4 rectangles)} = 0.25 \times (0.0625 + 0.25 + 0.5625 + 1) = 0.34375 \]

Key Concepts

Finite Approximation

Finite approximation is a technique used to estimate the area under a curve for a given function. Instead of calculating the exact area, which might be complex or impossible with certain functions, we use simple shapes such as rectangles. By dividing the area under the curve into several rectangles, we can get a closer approximation. This method becomes more precise as the number of rectangles increases.

In the exercise, we are interested in approximating the area under the curve of the function \(f(x) = x^2\) from \(x = 0\) to \(x = 1\). By dividing this interval into two or four segments, we create rectangles that form either a lower or upper sum approximation. The idea is to use measurable shapes to estimate the curve's area, accepting some error depending on the number of rectangles used.

This method of approximation highlights the importance of partitioning intervals correctly and choosing endpoints wisely to simulate the area as accurately as possible.

Upper and Lower Sums

Upper and lower sums are methods to approximate the area under a curve by using a series of rectangles. The choice of using either the lower or upper edge of a curve to define the rectangle's height determines the type of sum calculated.

• **Lower Sum:** This involves drawing rectangles under the curve, with the rectangle's height determined from the function value at the left endpoint of each interval. This approach potentially underestimates the area. In our exercise, for example, using the left endpoints at \(f(x)\) gives values like \(0\) and \(0.25\) for 2 rectangles, creating a lower bound to the area.


• **Upper Sum:** Here, rectangles are placed over the curve, using the right endpoint to determine the height. This typically overestimates the area. When we use values like \(0.25\) and \(1\) from the right endpoints for each rectangle in our example, the total area is expected to exceed the true value.

The sum of these areas becomes more accurate as the number of rectangles increases. Therefore, both upper and lower sums converge to the exact area with sufficiently small widths of partition.

Integral Calculus

Integral calculus deals with the computation of areas, volumes, and other concepts that arise through accumulation. It is fundamentally about finding function integrals, which represent the area under curves as precise, distinct values.

In the context of our exercise, we apply the idea of Riemann sums—finite approximations using upper and lower sums—to effectively practice integral calculus. These approximations become integrals in the limit as the number of rectangles approaches infinity.

When we partition the interval from \(x=0\) to \(x=1\) and increase the number of rectangles, the Riemann sum approaches the definite integral of the function \(f(x)=x^2\), symbolized by \(\int_0^1 x^2 \, dx\). This integral signifies the exact area under the curve, grasping the essence and power of integral calculus in analysis.

By using the concept of finite approximations and invoking the Riemann sum principles, students transition from visual numeric methods to understanding the more rigorous formulations of calculus.