Problem 54 - Biocalculus Calculus for the Life Sciences

Question

Suppose that a volcano is erupting and readings of the rate \(r(t)\) at which solid materials are spewed into the atmosphere are given in the table. The time \(t\) is measured in seconds and the units for \(r(t)\) are tonnes (metric tons) per second. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline r(t) & {2} & {10} & {24} & {36} & {46} & {54} & {60} \\\ \hline\end{array}$$ (a) Give upper and lower estimates for the total quantity \(Q(6)\) of erupted materials after 6 seconds. (b) Use the Midpoint Rule to estimate \(Q(6)\)

Step-by-Step Solution

Answer

Lower: 172 tonnes, Upper: 230 tonnes, Midpoint: 201 tonnes.

Step 1: Understand the Problem

We need to estimate the total erupted material over 6 seconds using provided rate data. First, let's analyze the data to understand how it might be used to get these estimates.

Step 2: Lower Estimate Using Left Endpoint Approximation

We will approximate the integral using left endpoint of each subinterval. To do this, we multiply each rate value (excluding the last one) by the change in time between consecutive readings, which is 1 second.\[Q_{lower} = (r(0) \times 1) + (r(1) \times 1) + (r(2) \times 1) + (r(3) \times 1) + (r(4) \times 1) + (r(5) \times 1)\]\[ = (2 \times 1) + (10 \times 1) + (24 \times 1) + (36 \times 1) + (46 \times 1) + (54 \times 1) = 172 \]Therefore, the lower estimate of the total quantity is 172 tonnes.

Step 3: Upper Estimate Using Right Endpoint Approximation

In this step, we use the right endpoint of each interval, which means starting with the second rate value and continuing until the last.\[Q_{upper} = (r(1) \times 1) + (r(2) \times 1) + (r(3) \times 1) + (r(4) \times 1) + (r(5) \times 1) + (r(6) \times 1)\]\[ = (10 \times 1) + (24 \times 1) + (36 \times 1) + (46 \times 1) + (54 \times 1) + (60 \times 1) = 230 \]Therefore, the upper estimate of the total quantity is 230 tonnes.

Step 4: Estimate Using the Midpoint Rule

For this estimate, we find the average rate in each interval and multiply by the interval length:\[Q_{midpoint} = (\frac{r(0) + r(1)}{2} \times 1) + (\frac{r(1) + r(2)}{2} \times 1) + (\frac{r(2) + r(3)}{2} \times 1) + (\frac{r(3) + r(4)}{2} \times 1) + (\frac{r(4) + r(5)}{2} \times 1) + (\frac{r(5) + r(6)}{2} \times 1) \]\[ = (\frac{2 + 10}{2} \times 1) + (\frac{10 + 24}{2} \times 1) + (\frac{24 + 36}{2} \times 1) + (\frac{36 + 46}{2} \times 1) + (\frac{46 + 54}{2} \times 1) + (\frac{54 + 60}{2} \times 1)\]\[ = (6) + (17) + (30) + (41) + (50) + (57) = 201 \]The estimated total quantity using the Midpoint Rule is 201 tonnes.

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical technique used to estimate the value of an integral, especially when calculating the total accumulation of a continuous rate like the one observed during a volcanic eruption. In simpler terms, it helps us estimate the total quantity of material erupted over a period.
The process involves dividing the time into intervals and using the midpoint of each interval for calculation. For example, with readings at every second from 0 to 6, the midpoints are calculated between each pair of consecutive time values.

• Calculate each midpoint: These are the values between consecutive readings, for instance, the midpoint between time 0 and 1 seconds is 0.5 seconds.
• Use these midpoints to estimate the average rate: This involves taking the function values at these midpoints.
• Multiply the average by the width of each interval: In our case, each interval is 1 second long.

Through this method, \[Q_{midpoint} = (6) + (17) + (30) + (41) + (50) + (57) = 201 \]This means using midpoints for calculation, you estimate a total of 201 tonnes of material erupted.

Upper and Lower Estimates

Upper and Lower Estimates create bounds when approximating an integral, which can be visualized as the total area under a rate curve. These bounds are essential because they provide a range within which the actual integral value lies.
The Lower Estimate uses the Left Endpoint Approximation:

• Choose the left value of each interval. For example, from 0 to 1 second, use the rate at 0 seconds.
• Multiply this rate by the interval width (1 second in this case).

Calculate for each interval:\[Q_{lower} = 172 \]This result rests at the lower boundary of the true value.
The Upper Estimate uses the Right Endpoint Approximation:

• Select the right value of each interval, meaning from 0 to 1 second, use the rate at 1 second, and so on.
• Again, multiply by the width of each interval.

For the upper estimate:\[Q_{upper} = 230 \]This result gives you the upper bound for the erupted material. Together, these estimates frame where the actual amount of erupted material will lie between 172 and 230 tonnes.

Estimation Techniques

Various techniques are used for numerical integration, and selecting the appropriate one can depend on the data and situation. Let's highlight the most common approaches, like the ones involved in this exercise:

• **Midpoint Rule:** Utilizes the midpoint of intervals for a more balanced estimate, ideal when reaction time and fluctuations occur evenly across intervals.
• **Left and Right Endpoint Approximations:** These use the start or end of intervals which might cause either underestimation or overestimation as they assume the rate remains constant based on endpoint readings.
• Diverse methodologies exist like Trapezoidal and Simpson’s Rule that further balance the estimation by simulating more realistic curves over sections.

These methods all aim to give an accurate number for evaluating the integral, depending on the setup's symmetrical nature or data reliability.

Integral Approximation

Integral Approximation is a fundamental mathematical concept that estimates total values from rate data. When precise functions cannot be used or are unavailable, we depend on approximation.
These approximations are essential in scenarios involving real-world data, like volcanic eruptions where actual function data isn't precisely calculable. Estimation methods help fill this gap.

• Approximation forms include numerical integration techniques like those discussed, converting discrete data points into continuous estimates.
• Monte Carlo and Riemann Sum techniques are further robust strategies used across fields like physics, engineering, and anywhere interim data points exist.

By approximating the integral, we gauge how much material erupts by estimating the area under the rate versus time graph, thus translating to real-world phenomena understanding.