Problem 40 - Applied Calculus

Question

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=e^{x / 3} \text { from } x=0 \text { to } x=3 $$

Step-by-Step Solution

Answer

The area under the curve is \( 3e - 3 \).

Step 1: Understanding the Problem

We need to find the area under the curve defined by the function \( f(x) = e^{x/3} \) from \( x = 0 \) to \( x = 3 \). This area can be calculated using a definite integral.

Step 2: Setting Up the Integral

To find the area under the curve from \( x = 0 \) to \( x = 3 \), we set up the definite integral \( \int_{0}^{3} e^{x/3} \, dx \).

Step 3: Integrating the Function

The integral of \( e^{ax} \) is \( \frac{1}{a}e^{ax} \). In this case, \( a = \frac{1}{3} \), so the integral of \( e^{x/3} \) becomes \( 3e^{x/3} \).

Step 4: Evaluating the Definite Integral

Evaluate \( 3e^{x/3} \) from \( x = 0 \) to \( x = 3 \):\[ \left[ 3e^{x/3} \right]_{0}^{3} = 3e^{3/3} - 3e^{0} = 3e - 3 \]

Step 5: Simplifying the Expression

The expression \( 3e - 3 \) cannot be simplified further, so it represents the exact area under the curve from \( x = 0 \) to \( x = 3 \).

Step 6: Sketching the Curve

The curve of \( f(x) = e^{x/3} \) is an upward-slowing exponential curve starting from \( x=0 \) to \( x=3 \). The area under this curve between these points looks like a curved shape resting above the x-axis between these values.

Key Concepts

Area Under a Curve

The concept of the "area under a curve" is crucial in understanding integrals and their real-world applications. When we talk about finding the area under a function such as \( f(x) = e^{x/3} \), we're looking at a method to calculate the total space between the curve of the function and the x-axis between two specific points, in this case, from \( x = 0 \) to \( x = 3 \).

This calculation is done using a definite integral, which acts as a summative tool to add up infinitesimally small slices of area under the curve. For example, if you were stacking lots of thin paper strips under this curve, integrating would tell you their total thickness. The definite integral \( \int_{0}^{3} e^{x/3} \, dx \) precisely accomplishes this. The result gives us a number, which represents the total area laden beneath the curve over the specified interval.

Exponential Function

Understanding the nature of an "exponential function" is key to working with integrals involving these functions. Exponential functions like \( e^{x/3} \) grow at a rate proportional to their current value, which means they increase quite rapidly.

Features of exponential functions include:

• A rapid rise or decay based on the exponent's sign and base value.
• The base \( e \) is approximately 2.718 and represents natural growth or decay processes.
• Exponential functions dominate scenarios involving continuous growth, like population dynamics, radioactive decay, and interest calculations.

In our example, the function \( f(x) = e^{x/3} \) represents a smoother, slow-growth curve due to the division of \( x \) by 3. It exhibits the typical exponential growth behavior without being too steep, providing a moderate increase from \( x = 0 \) to \( x = 3 \).

Definite Integral Evaluation

"Definite integral evaluation" is the process of calculating the specific value of an integral over a given interval. This involves integrating the function over the defined limits and then computing the resultant area.

To evaluate \( \int_{0}^{3} e^{x/3} \, dx \), we first process the general rule for integrating an exponential function. The integral of \( e^{ax} \) is \( \frac{1}{a} e^{ax} \), where \( a \) is a constant. Hence, \( e^{x/3} \) integrates to \( 3e^{x/3} \) since \( a = \frac{1}{3} \).

Next, plug in the upper and lower limits of the interval into the integrated function:

• Calculate \( 3e^{3/3} = 3e \)
• Subtract \( 3e^{0} = 3 \)
• The answer is \( 3e - 3 \), which represents the exact area.

This value denotes the total area enclosed between the exponential curve and the x-axis from \( x = 0 \) to \( x = 3 \). The calculated area helps visualize and quantify aspects like growth over time, contributing to a deeper understanding of many natural and theoretical phenomena.