Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

The exact area of the region bounded by the graphs of $$y=\sqrt{t} \quad y=0 \quad \text { and } \quad t=5 $$ is \(\frac{2}{3} 5^{3 / 2} \doteq 7.453560\). Compute the upper rectangular approximation and the trapezoidal approximation to the area based on 50 subintervals and compute their relative accuracies.

Step-by-Step Solution

Answer

Use URAM and the trapezoidal rule with 50 subintervals to approximate and evaluate relative accuracy against \(\frac{2}{3}5^{3/2}\).

Step 1: Understanding the Problem

The problem requires us to approximate the area under the curve of the function \( y = \sqrt{t} \) from \( t = 0 \) to \( t = 5 \) using two methods: the Upper Rectangular Approximation Method (URAM) and the Trapezoidal Rule, with 50 subintervals. Furthermore, we need to find their relative accuracies compared to the exact area, \( \frac{2}{3} 5^{3/2} \).

Step 2: Calculate the Exact Area

The exact area is given by \( A = \frac{2}{3} \times 5^{3/2} \). First, calculate \( 5^{3/2} = (\sqrt{5})^3 \). Since \( \sqrt{5} \approx 2.236 \), then \( (2.236)^3 \approx 11.180 \). Therefore, \( A = \frac{2}{3} \times 11.180 \approx 7.453 \).

Step 3: Divide the Interval into Subintervals

The interval is from \( t = 0 \) to \( t = 5 \), and we divide it into 50 equal subintervals. Each subinterval has a width \( \Delta t = \frac{5 - 0}{50} = 0.1 \).

Step 4: Calculate Upper Rectangular Approximation

For URAM, use the right endpoints of the subintervals. Calculate \( y = \sqrt{t} \) at each right endpoint from \( t = 0.1 \) to \( t = 5 \). Sum these heights and multiply by \( \Delta t = 0.1 \). The approximation \( A_{UR} = 0.1 \times \left( \sqrt{0.1} + \sqrt{0.2} + \cdots + \sqrt{5} \right) \).

Step 5: Calculate Trapezoidal Approximation

For the trapezoidal rule, average the function values at the endpoints of each interval. This gives \( A_T = 0.1 \times \left( \frac{\sqrt{0} + \sqrt{0.1}}{2} + \frac{\sqrt{0.1} + \sqrt{0.2}}{2} + \ldots + \frac{\sqrt{4.9} + \sqrt{5}}{2} \right) \).

Step 6: Calculate Relative Accuracies

First, compute any approximation values obtained. The relative accuracy of each method is calculated as \( R = \left| \frac{A_{approx} - A}{A} \right| \times 100\% \), where \( A \) is the exact area. Compute \( R_{UR} \) and \( R_T \) for the URAM and trapezoidal approximations respectively.

Key Concepts

Upper Rectangular Approximation

The Upper Rectangular Approximation Method (URAM) is a technique used to estimate the area under a curve on a graph. It is particularly useful when dealing with integrals that are difficult to solve analytically.

In URAM, we approximate the area by summing up the areas of rectangles. Here, each rectangle's upper right corner touches the curve, hence the term "upper." For the function \( y = \sqrt{t} \), and the interval from \( t = 0 \) to \( t = 5 \), we divide the interval into 50 smaller subintervals. The width of each subinterval, \( \Delta t \), is calculated as \( \frac{5-0}{50} = 0.1 \).

The height of each rectangle comes from the function value at the right endpoint of each subinterval. For example, the height corresponding to the subinterval from \( 0.1 \) to \( 0.2 \) would be \( \sqrt{0.2} \).

To find the total approximate area \( A_{UR} \) under the curve, you sum up all the heights and multiply by the width:

• \( A_{UR} = 0.1 \times (\sqrt{0.1} + \sqrt{0.2} + \ldots + \sqrt{5}) \)

Trapezoidal Rule

The Trapezoidal Rule is another method for estimating the area under a curve. This method is often considered more accurate than the rectangle methods, like URAM, because it takes into account the slope of the curve by essentially "connecting the dots" between two points with a straight line.

For the same interval (\( 0 \) to \( 5 \)) and the function \( y = \sqrt{t} \), we again divide the interval into 50 subintervals. However, unlike URAM, we now use the values at both endpoints of each subinterval to create a series of trapezoids rather than rectangles.

The area of each little trapezoid is calculated by taking the average of the two heights (i.e., the function values at the endpoints) and then multiplying by the width \( \Delta t \):

• \( A_T = 0.1 \times \left( \frac{\sqrt{0} + \sqrt{0.1}}{2} + \frac{\sqrt{0.1} + \sqrt{0.2}}{2} + \ldots + \frac{\sqrt{4.9} + \sqrt{5}}{2} \right) \)

This sequential averaging process accounts for changes in the slope of the curve, giving an approximation that tends to converge faster to the exact value than simple rectangular methods.

Relative Accuracy

When approximating the area under a curve, it is important to understand how close these approximations (URAM and Trapezoidal Rule) are to the actual area. This is where the concept of Relative Accuracy comes into play.

Relative accuracy is a measure of the error between the approximate value (whether URAM or Trapezoidal) and the exact known area. The expression for relative accuracy is given by:

\[ R = \left| \frac{A_{approx} - A}{A} \right| \times 100\% \]

Here, \( A \) represents the exact area under the curve, which for this problem is \( \frac{2}{3} 5^{3/2} \approx 7.453560 \). \( A_{approx} \) would be either \( A_{UR} \) or \( A_T \).

By calculating \( R \), one can quantify the accuracy of each method as a percentage. This helps assess how well either approximation method performed compared to the known exact value. High relative accuracy indicates that the approximation is close to the exact value, which is crucial for validating the efficiency and reliability of numerical methods in practice.