Problem 67 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the area of the region bounded by the curves \(y=3 e^{-x / 3}, y=0, x=0\), and \(x=9\). Make a sketch.

Step-by-Step Solution

Answer

The area of the region is approximately 8.55 square units.

Step 1: Understanding the Problem

To find the area of the region bounded by the curves, we need to focus on the given boundaries: the curve defined by \(y = 3e^{-x/3}\), the line \(y = 0\) (the x-axis), and the vertical lines \(x = 0\) and \(x = 9\). We are essentially finding the area under the curve \(y = 3e^{-x/3}\) from \(x = 0\) to \(x = 9\).

Step 2: Setting Up the Integral

The area under a curve \(y = f(x)\) from \(x = a\) to \(x = b\) is given by the definite integral \(\int_{a}^{b} f(x) \, dx\). In our case, \(f(x) = 3e^{-x/3}\), \(a = 0\), and \(b = 9\). So, the area is given by \(\int_{0}^{9} 3e^{-x/3} \, dx\).

Step 3: Calculating the Integral

To solve the integral \(\int 3e^{-x/3} \, dx\), we perform a substitution. Let \(u = -x/3\), then \(du = -1/3 \, dx\) or equivalently \(dx = -3 \, du\). The integral becomes \(-9 \int e^{u} \, du\). The integral of \(e^u\) is \(e^u\), so we have \(-9e^u\). Substituting back for \(u\), we get \(-9e^{-x/3}\).

Step 4: Applying the Limits

Evaluate the integral from \(x = 0\) to \(x = 9\): \([-9e^{-x/3}]_{0}^{9}\). This becomes \(-9e^{-9/3} - (-9e^{0})\). Calculating this gives \(-9e^{-3} + 9\).

Step 5: Simplifying the Expression

Simplify \(-9e^{-3} + 9\). Note that \(e^{-3}\) is a constant, approximately 0.0498. Therefore the expression simplifies to \(9 - 9(0.0498)\). This equates to \(9 - 0.4482 = 8.5518\).

Step 6: Final Answer and Sketch

The area of the region is approximately 8.55 square units. The sketch would show a decaying exponential curve starting at \(y = 3\) and approaching the x-axis, bounded between \(x = 0\) and \(x = 9\).

Key Concepts

Area under a Curve

The "area under a curve" refers to the space between the curve itself and the x-axis over a certain interval. To find this area, we use integration, a key concept in calculus.

In mathematical terms, if you have a function \( y = f(x) \), the area between this curve and the x-axis from \( x = a \) to \( x = b \) is calculated using the definite integral \( \int_{a}^{b} f(x) \, dx \). This integral processes a continuous sum of the infinite rectangular strips under the curve and provides the area.

To understand integration further, imagine cutting the space under the curve into very thin vertical strips, then summing the area of these strips. This approach would exactly give the area under the curve when the strips are thin enough, which is what integration effectively does.

• Integration allows us to calculate not just the total area but also areas defined by certain bounds.
• It is an incredibly powerful tool as it consolidates the contributions of every infinitesimal part under the curve.

Exponential Function

An exponential function is a type of mathematical function that involves an exponent, which is the power to which a number or expression (called the base) is raised. For example, in the exercise, the curve is described by the exponential function \( y = 3e^{-x/3} \).

In this function, \( e \) is the base, known as Euler's number, which is approximately equal to 2.71828. It is a crucial constant in mathematics due to its unique properties, especially in calculus. Here's why exponential functions like our example are significant:

• The base \( e \) ensures that the rate of growth or decay is proportional to the current value, which is important in natural growth processes.
• The function \( 3e^{-x/3} \) is a decaying function because of the negative exponent \(-x/3\). This negative sign means as x increases, the value of the function decreases, which is why the curve eventually approaches the x-axis.
• Understanding the structure of exponential functions helps in visualizing and sketching the curve correctly.

Bounded Region

The concept of a "bounded region" involves a specific area enclosed within certain limits or boundaries on a graph. In this exercise, the region is bounded by various mathematical expressions:

• The curve \( y = 3e^{-x/3} \),
• The x-axis \( y = 0 \),
• The vertical lines \( x = 0 \) and \( x = 9 \).

These boundaries define the limits of the area we are interested in calculating.

Visualizing a bounded region can be likened to drawing a fenced area on your graph paper, where the lines and curves act as the fences. This "fenced" area is the only region taken into account when determining the area under a curve using a definite integral.

An important point to understand is that a region must be closed by boundaries to be measured for its area effectively. Double-checking the boundaries ensures accurate integration limits, ensuring no part of the region goes unnoticed in your calculations.