Problem 8 - Applied Calculus

Question

The following table gives world oil consumption, in billions of barrels per year. ' Estimate total oil consumption during this 25 -year period. $$ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 1980 & 1985 & 1990 & 1995 & 2000 & 2005 \\ \hline \text { Oil (bn barrels/yr) } & 22.3 & 21.3 & 23.9 & 24.9 & 27.0 & 29.3 \\\ \hline \end{array} $$

Step-by-Step Solution

Answer

Total consumption is estimated to be 614 billion barrels.

Step 1: Understanding the Problem

We need to estimate the total oil consumption over a 25-year period based on the given annual consumption values for certain years. The given years are 1980, 1985, 1990, 1995, 2000, 2005, and we must find an estimation method to calculate the consumption over the intervals between these years.

Step 2: Choosing an Estimation Method

We'll use the trapezoidal rule to estimate the total oil consumption over this period. This method approximates the area under the consumption curve as a series of trapezoids, providing a reasonable estimation for integral calculations over intervals where the start and end values are known.

Step 3: Calculate Each Interval Contribution

Using the trapezoidal rule, calculate the oil consumption for each interval. The formula for the trapezoid area is given by \(A = \frac{1}{2} \times (b_1 + b_2) \times h\)where \(b_1\) and \(b_2\) are the values at the endpoints of the interval, and \(h\) is the time period, in this case, 5 years.

Step 4: Interval 1980 to 1985

Calculate the consumption difference:\[ A_1 = \frac{1}{2} \times (22.3 + 21.3) \times 5 = 108 \, \text{billion barrels} \]

Step 5: Interval 1985 to 1990

Calculate the consumption difference:\[ A_2 = \frac{1}{2} \times (21.3 + 23.9) \times 5 = 113 \, \text{billion barrels} \]

Step 6: Interval 1990 to 1995

Calculate the consumption difference:\[ A_3 = \frac{1}{2} \times (23.9 + 24.9) \times 5 = 122 \, \text{billion barrels} \]

Step 7: Interval 1995 to 2000

Calculate the consumption difference:\[ A_4 = \frac{1}{2} \times (24.9 + 27.0) \times 5 = 129.75 \, \text{billion barrels} \]

Step 8: Interval 2000 to 2005

Calculate the consumption difference:\[ A_5 = \frac{1}{2} \times (27.0 + 29.3) \times 5 = 141.25 \, \text{billion barrels} \]

Step 9: Summing Up the Contributions

Sum all the interval contributions to get the total consumption over the 25-year period: \[\text{Total} = A_1 + A_2 + A_3 + A_4 + A_5 = 108 + 113 + 122 + 129.75 + 141.25 = 614 \, \text{billion barrels} \]

Step 10: Providing the Final Answer

Based on the trapezoidal rule calculation, the estimated total oil consumption over the 25-year period is 614 billion barrels.

Key Concepts

Oil Consumption Estimation

Estimating oil consumption is crucial for understanding global energy usage over specific time periods. In our exercise, we aim to estimate the total oil consumption over a span of 25 years, using data provided for specific years within that timeframe. The task here is to find a way to effectively utilize the available consumption data points from 1980 through 2005.

The estimation method is key to deriving a meaningful result. Since we don't have continuous data but only values at discrete points, we employ mathematical techniques to approximate what happens between those points. For this, the Trapezoidal Rule is applied, as it helps us to estimate the total consumption by approximating the area under the consumption curve over a series of intervals.

This technique allows us to use the given figures from each specified year, enabling us to estimate the consumption that occurred during the intervals between these years. By calculating these intermediate values, we achieve a comprehensive estimation of oil consumption across the 25-year period.

Area Under Curve

The 'area under the curve' is a key concept in understanding integral calculations, especially in approximation contexts like this exercise. When we talk about finding the area under a curve, we're essentially referring to the total accumulated quantity. In this case, that quantity is oil consumption over the years.

The consumption values given represent years plotted against the amount of oil consumed annually. If we were to graph these points, they'd form a curve outlining oil consumption trends over time. Calculating the area under this curve helps us understand the total oil consumed during the given time period.

Utilizing the Trapezoidal Rule simplifies our task. Instead of needing precise data and complex calculus calculations, we can form trapezoids between the points for each five-year interval. By finding the area of each trapezoid and summing them up, we determine the approximate area under the full curve, representing the total oil consumption for the 25-year span.

Integral Calculations

Integral calculations help in determining the area under a curve, representing total quantities over time such as oil consumption. While exact integration requires complex computations, approximation methods like the Trapezoidal Rule simplify the process using simple geometry.

The trapezoidal rule is based on creating trapezoids between successive points. To calculate the area of each trapezoid, we use the formula: \[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]where \(b_1\) and \(b_2\) are the base values at the endpoints of the interval, representing oil consumption in billion barrels per year, and \(h\) is the interval duration, in this case, 5 years.

This method involves breaking down the curve into manageable sections and approximating the integral piece by piece. By summing up these individual contributions, we arrive at an estimated total consumption figure, highlighting the power of integral approximations in real-world scenarios.