Question

U.S. Oil Consumption The rate of consumption of oil in the United States during the 1980 s (in billions of barrels per year) is modeled by the function \(C=27.08 \cdot e^{t / 25}\) , where \(t\) is the number of years after January \(1,1980 .\) Find the total consumption of oil in the United States from January \(1,1980\) to January 1 , 1990.

Short answer

The total consumption of oil in the billions of barrels is found by adding up the two evaluated integral terms

Step-by-step solution

Identify the integral to be solved

To find the total quantity consumed, the given rate function should be integrated over the given period. So we need to evaluate \( \int_{0}^{10} 27.08 \cdot e^{t / 25} dt \)

Solve the integral

Use the basic integral property \( \int e^x dx = e^x + C \) to find the integral. Integrate the function by parts as per the expression \( \int u dv = uv - \int v du\), taking \(u = e^{t / 25}\) and \(dv = 27.08 dt\); this leads to \(du = e^{t / 25}/25 dt\) and \(v = 27.08t\). Then the integral becomes \( \int_{0}^{10} u dv = \int_{0}^{10} e^{t / 25} 27.08 dt = [u v]_{0}^{10}- \int_{0}^{10} v du\). This simplifies to \( [27.08t e^{t / 25}]_{0}^{10}- 27.08 \int_{0}^{10} t e^{t / 25}dt\). The second term here can be evaluated using integration by parts again or by using an integral table.

Evaluation of terms and summing up

Evaluate each term at the upper and lower bounds, and sum the two terms to find the total oil consumed in the period from t=0 to t=10. The evaluated integral will give the answer in billions of barrels.

Key concepts

Indefinite Integration

Indefinite integration is a fundamental concept that you'll come across in calculus. It's the process of finding the original function given its derivative, which is often referred to as its antiderivative. It's called 'indefinite' because, unlike definite integration, it does not refer to a specific interval and therefore the result includes an arbitrary constant (usually denoted as +C) since there are infinitely many antiderivatives for any function.

For example, if you're given the derivative as an exponential function like \( e^{x} \), the indefinite integral would be \( \int e^{x} dx = e^{x} + C \). Indefinite integration is the stepping stone towards understanding advanced integration techniques and applications, from physics to finance.

Integration by Parts

Integration by parts is one of the essential techniques in integral calculus, especially when you encounter products of functions in integration. This method is based on the product rule for derivatives and is formally given by \( \int u dv = uv - \int v du \).

The strategy is to choose \( u \) and \( dv \) wisely from the original integrand so that \( du \) and \( v \) are easy to work with. In real-world scenarios, like calculating the total growth over time when a rate is changing, integration by parts can be essential. For example, when integrating an exponential function multiplied by a polynomial, splitting the terms into \( u \) and \( dv \) can simplify the problem significantly.

Exponential Functions

Exponential functions are ubiquitous in the real world, defining processes that grow or decay at rates proportional to their current value. Defined by the equation \( f(x) = a \cdot e^{kx} \), where \( e \) is Euler's number, \( a \) is a constant, and \( k \) is the rate of growth or decay, these functions model phenomena ranging from compound interest to radioactive decay.

For the U.S. Oil Consumption problem, the function \( C=27.08 \cdot e^{t / 25} \) represents the rate of consumption of oil: it's an exponential function where the rate of consumption is proportional to the current level of consumption. Understanding how to integrate such functions is crucial in predicting total quantities over time.

Application of Integrals in Economics

Integration finds immense application in the field of economics, providing tools to deal with continuous growth models and aggregate quantities. For instance, when we talk about the total consumption of a commodity, like oil, we are discussing an integral of a rate function over a certain period.

By integrating the rate function from the start to the end of a period, one can quantify total consumption, production, or investment. In the exercise given, the integral calculates the total oil consumption over a decade by integrating the consumption rate, which is an exponential function indicating growth. This type of analysis is critical for forecasting, planning, and policy-making in economics.