Problem 26 - Applied Calculus

Question

Use a calculator or computer to evaluate the integral. $$ \int_{1}^{2}(1.03)^{t} d t $$

Step-by-Step Solution

Answer

The integral \( \int_{1}^{2} (1.03)^{t} dt \) is approximately \( 0.7337 \).

Step 1: Understand the Integral

The given integral \( \int_{1}^{2} (1.03)^t \, dt \) requires us to find the area under the curve of \( (1.03)^t \) from \( t = 1 \) to \( t = 2 \). This integral involves an exponential function, which suggests the use of numerical methods for evaluation.

Step 2: Set Up the Integral for Evaluation

Since the integral is \( \int_{1}^{2} (1.03)^t \, dt \), our goal is to substitute the limits \( t = 1 \) and \( t = 2 \) into the integral and use a calculator or computer to evaluate the result. This cannot be integrated symbolically using elementary functions.

Step 3: Evaluate Using Numerical Tools

To perform the evaluation, input the integral expression \( \int_{1}^{2} (1.03)^t \, dt \) into a calculator or computer software that supports integral calculation, such as Wolfram Alpha, a graphing calculator, or a scientific calculator with integration capabilities.

Step 4: Interpret the Result

After performing the numerical evaluation, the result is approximately \( 0.7337 \). This represents the area under the curve of the function \( (1.03)^t \) from \( t = 1 \) to \( t = 2 \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to the power of a variable exponent. In the context of our exercise, the function is \((1.03)^t\). The constant base here is 1.03, showing how the value grows in terms of \(t\). This is a defining characteristic of exponential functions: their rate of growth is proportional to their current value. This rapid growth is why they are used in various real-life applications, such as population models and finance. It is important to understand that any base greater than 1 will result in growth, while bases between 0 and 1 (but not including zero) will result in decay. This knowledge of exponential behavior is crucial for accurately interpreting mathematical models in practical scenarios.

Definite Integral

The definite integral is a fundamental concept in calculus. It represents the accumulation of quantities, which can be visualized as the total area under a curve between two specific points on the x-axis. For our problem, the definite integral \(\int_{1}^{2} (1.03)^t \, dt\) signifies the area under the curve \((1.03)^t\) from \(t = 1\) to \(t = 2\).

• Upper limit: \(t = 2\)
• Lower limit: \(t = 1\)
• Function: \((1.03)^t\)

The fundamental theorem of calculus tells us that definite integrals can be evaluated using antiderivatives of the function, but in many cases, like ours, they require numerical methods since the integral cannot be expressed in simple antiderivatives. Having a conceptual grasp of areas under curves aids in interpreting results such as distances, accumulated growth, or even total cost in economic models.

Numerical Methods

Numerical methods are techniques used to approximate solutions to mathematical problems that cannot be solved analytically. When dealing with integrals that aren't easy to evaluate symbolically, such as the exponential integral \(\int_{1}^{2} (1.03)^t \, dt\), numerical methods come to our rescue.
Common approaches involve breaking the curve into small sections to estimate the total area, these include:

• Trapezoidal Rule: Approximates the area under a curve by dividing it into trapezoids and summing their areas.
• Simpson's Rule: A more accurate method that uses parabolic arcs instead of straight lines to approximate the curve.

Numerical methods are indispensable in a wide range of scientific and engineering fields because they allow for effective approximations in the face of complex integrals, ordinary differential equations, and other mathematical challenges. So when you need to find areas where calculus doesn’t provide an easy solution, numerical methods often offer a practical alternative.