Problem 9 - 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

The trapezoidal approximation is generally more accurate than the upper rectangular approximation for the given problem.

Step 1: Determine the Interval

The function provided is \( y = \sqrt{t} \). The interval for \( t \) over which we evaluate the area is from \( t=0 \) to \( t=5 \). This defines our bounds for the area calculation.

Step 2: Set up the Upper Rectangular Approximation

For the Upper Rectangular Approximation with 50 subintervals, calculate the width of each subinterval as \( \Delta t = \frac{5 - 0}{50} = 0.1 \). The upper rectangular approximation uses the right endpoint of each subinterval to estimate area, i.e., \( A_{URA} = \sum_{i=1}^{50} f(t_i) \Delta t \), where \( t_i = i \times 0.1 \).

Step 3: Calculate the Upper Rectangular Approximation

Substitute \( f(t) = \sqrt{t} \) into the formula: \( A_{URA} = \sum_{i=1}^{50} \sqrt{i \times 0.1} \times 0.1 \). This requires summing over the square roots of \( 0.1, 0.2, ..., 5 \), scaled by 0.1. Compute this sum numerically to obtain the approximation.

Step 4: Set up the Trapezoidal Approximation

The Trapezoidal Approximation is given by \( A_{TA} = \frac{\Delta t}{2} \left( f(t_0) + 2 \sum_{i=1}^{49} f(t_i) + f(t_{50}) \right) \) with \( \Delta t = 0.1 \). Calculate \( f(t) = \sqrt{t} \) at each subinterval point.

Step 5: Calculate the Trapezoidal Approximation

Compute \( A_{TA} = \frac{0.1}{2} \times (\sqrt{0} + 2 \times (\sum_{i=1}^{49} \sqrt{i \times 0.1}) + \sqrt{5}) \). Perform the sum and multiply by 0.05 to find the trapezoidal approximation.

Step 6: Compare Approximations to Exact Value

The exact area is \( \frac{2}{3} \times 5^{3/2} \approx 7.453560 \). Calculate the relative accuracy of each approximation: \( \text{Accuracy} = \left( 1 - \frac{|\text{Approximation} - \text{Exact}|}{|\text{Exact}|} \right) \times 100\% \). Do this for both the Upper Rectangular Approximation and the Trapezoidal Approximation.

Step 7: Final Results

After computing all sums and comparing them to the exact area, you'll find that the trapezoidal approximation is usually closer to the exact area due to its method of averaging the function values at the interval endpoints, which compensates better for curve shape. Calculate precise values for comparison.

Key Concepts

Rectangular Approximation Method

The Rectangular Approximation Method is a technique used in calculus to estimate the area under a curve. This area can be thought of as representing a definite integral. The method involves approximating a curve with a series of rectangles. The height of each rectangle is determined by the function value at either the left endpoint, midpoint, or right endpoint of each subinterval.

In the case of the Upper Rectangular Approximation, we use the right endpoint of each subinterval to compute the height. This can sometimes lead to an overestimation or underestimation depending on whether the function is increasing or decreasing. For example, with the function \( y = \sqrt{t} \) from \( t=0 \) to \( t=5 \), we split this interval into 50 subintervals. The width of each subinterval, \( \Delta t \), is \( 0.1 \) as calculated by \( \frac{5 - 0}{50} \).

To find the area using this method, the formula \( A_{URA} = \sum_{i=1}^{50} f(t_i) \Delta t \) is used. Here, \( f(t_i) = \sqrt{i \times 0.1} \), and each small rectangle's area is summed to get an overall approximation of the total area. This method is quite straightforward and gives a clear visual approach by partitioning the region into simple geometric shapes.

Trapezoidal Approximation

The Trapezoidal Approximation is another method used for estimating the area under a curve, which again represents a definite integral. This approach uses trapezoids instead of rectangles, which more closely follows the curve's shape by taking the average of the function values at the endpoints of each subinterval.

For the function \( y = \sqrt{t} \) over the interval \( t=0 \) to \( t=5 \), we again use 50 subintervals, just as with the rectangular approximation. The width of each subinterval is \( \Delta t = 0.1 \).

The formula for this approximation is \( A_{TA} = \frac{\Delta t}{2} \left( f(t_0) + 2 \sum_{i=1}^{49} f(t_i) + f(t_{50}) \right) \). Here, it calculates the area of each "trapezoid" formed by the curve and the interval limits. The sum gathers all these individual areas, offering a more refined approximation, particularly effective for curves that are relatively smooth.

The trapezoidal method is generally more accurate than the rectangular approximation because it better approximates the underlying curve, taking into account both endpoints of each partition and effectively smoothing out slight ups and downs of the function.

Definite Integrals

Definite integrals are a fundamental concept in calculus used to compute the accumulation of quantities, which can be represented as the total area under a curve between two points. The idea behind definite integrals is to sum an infinite number of infinitely small quantities, specifically by integrating a function between limits \( a \) and \( b \).

In mathematical terms, the definite integral of a function \( f(x) \) from \( a \) to \( b \) is denoted \( \int_{a}^{b} f(x) \, dx \). This yields the net area between the function and the x-axis—in our exercise example, the area underneath the curve \( y = \sqrt{t} \) from \( t=0 \) to \( t=5 \) represents such a definite integral, calculated exactly to be \( \frac{2}{3} \times 5^{3/2} \).

Definite integrals facilitate the understanding of how areas, distances, volumes, and other continuous accumulations can be determined practically and efficiently using mathematical formulas. The approximations like the rectangular and trapezoidal methods are closely related, providing numerical strategies to estimate integrals when an analytical solution might be complex or unobtainable.