Question

Sketch the region \(R\) bounded by the graphs of the given equations, and show a typical vertical slice. Then find the volume of the solid generated by revolving \(R\) about the \(x\) -axis. \(y=e^{x}, y=\frac{e}{x}, y=0\), between \(x=0\) and \(x=3\)

Short answer

The total volume is the sum of calculated integrals over the intervals.

Step-by-step solution

Sketch the Region Bounded by the Curves

First, consider the equations given for the region:- The curve given by the equation \(y = e^x\) is an exponential function that rises steeply with increasing \(x\).- The curve \(y = \frac{e}{x}\) is a hyperbola that decreases as \(x\) increases within the given interval.- The line \(y = 0\) represents the x-axis.Draw the coordinate system and plot these two curves, noting the behavior of \(y = e^x\) which starts at \(y = 1\) when \(x = 0\) and increases rapidly, and \(y = \frac{e}{x}\), which starts at \(y = e\) when \(x = 1\) and descends as \(x\) goes from 1 to 3. The region of interest, \(R\), is between these curves above the x-axis from \(x = 0\) to \(x = 3\).

Identify the Typical Vertical Slice

Select a representative vertical slice (strip) within the region \(R\). This slice is parallel to the y-axis, extending from the x-axis \(y = 0\) to the curve \(y = e^x\), minus the area below \(y = \frac{e}{x}\) and \(y = e^x\). For each strip, the width is an infinitesimally small segment \(\Delta x\), and the length changes along the x-axis, with the top of the strip being the lower of \(e^x\) and \(\frac{e}{x}\), and the bottom part \(y = 0\).

Set Up the Integral for Volume Using the Washer Method

Since we are revolving around the x-axis, calculate the volume using the Washer Method.The volume of each washer shaped by the strip is defined by:\[ V = \pi \left( (R_{ ext{outer}})^2 - (R_{ ext{inner}})^2 \right) dx \]Where:- \(R_{ ext{outer}} = \text{max}(e^x, \frac{e}{x})\) as we revolve, and both will be different over different intervals.- \(R_{ ext{inner}} = 0\) since we are using the x-axis \(y = 0\).

Calculate the Volume Across Specified Intervals

To find the function that defines each section of the washers, partition into intervals as necessary based on which function \(e^x\) or \(\frac{e}{x}\) is greater. Solve for the intersection points if needed to get clear partitions.Calculate the volume for each relevant interval using the total volume integral:1. From \(x=0\) to the intersection point \(x=1\): \[ V_1 = \pi \int_{0}^{1} \left((e^x)^2 - (\frac{e}{x})^2 \right) dx\]2. From \(x=1\) to \(x=3\): \[ V_2 = \pi \int_{1}^{3} \left((\frac{e}{x})^2 - 0^2 \right) dx\]Evaluate each integral separately.

Solve the Integrals and Sum the Volumes

Perform the integration:- For \(x=0\) to \(x=1\): \[ V_1 = \pi \left[ \frac{e^{2x}}{2} \right]_0^1 - \pi \left[ e^2 \ln|x| \right]_0^1 \] Resolve and compute this integral. - For \(x=1\) to \(x=3\): \[ V_2 = \pi \int_{1}^{3} \frac{e^2}{x^2} dx = \pi e^2 \left[ -\frac{1}{x} \right]_1^3\] Calculate this integral.Add the results of both volumes to get the final volume of the solid.

Key concepts

Washer Method

When calculating the volume of a solid of revolution, the washer method is a popular approach. This method is perfect for cases where the solid has a hole or a cavity inside. Imagine a washer, similar to those used in hardware, which is essentially a disc with a hole in the center. In calculus, to find the volume of such a solid, you revolve a region around an axis, commonly the x-axis or y-axis.

Here's the basic idea:

• Identify an outer radius, which is the distance from the axis of rotation to the outer edge of the region.
• Determine the inner radius, which is the distance from the axis to the inner edge (if any) of the region.
• Compute the volume of each differential washer: The outer cylinder minus the inner cylinder generates the washer shape.

To calculate the volume, use the integral formula:

• \[ V = \pi \int_{a}^{b} \left( R_{\text{outer}}^2 - R_{\text{inner}}^2 \right) \, dx \]

By integrating over the desired interval, you sum all tiny washers' volumes to get the total solid volume.

Integral Calculus

Integral calculus serves as an essential tool to determine the accumulation of quantities, such as areas under curves and, in this case, volumes of solids. It's basically about finding the total value from known point values, a process called 'integration'.

The idea of integration can be understood through several components:

• Definite Integral: Represents the accumulation of quantities across a specified interval \([a, b]\). It is expressed as: \[ \int_{a}^{b} f(x) \, dx \]
• Basic Techniques: Integration by substitution and integration by parts are common techniques.
• Fundamental Theorem of Calculus: Connects differentiation and integration, indicating that they are inverse processes.

When calculating the volume of a solid formed by revolving a region around an axis, setting up the correct integral is critical. One must carefully evaluate the limits of integration — be it a region's bounds or intersection points — and the function being integrated.

Exponential Function

The exponential function is a crucial mathematical concept characterized by the constant rise of the function value. Often written as \( e^x \), where \( e \) is Euler's number approximately valued at 2.718. This function is unique because its rate of growth is directly proportional to its current value, resulting in continuous rapid growth.

Key aspects of exponential functions to note:

• The initial value is typically 1 when \( x = 0 \), given by \( e^0 = 1 \).
• Exponential functions are always positive and increase on the interval \( x > 0 \).
• They have a horizontal asymptote along the x-axis, meaning the function approaches but never actually reaches zero as \( x \) becomes increasingly negative.

In the context of revolving volumes, the exponential curve can serve as either an outer or inner boundary, depending on its position compared to other functions in the problem.

Hyperbolic Function

Hyperbolic functions offer another dimension to solving calculus problems, often showing similarities with the more familiar trigonometric functions. They are expressed in the forms of hyperbolas and are used to describe shapes and phenomena inherent to geometry and calculus, such as \( y = \frac{e}{x} \) in this problem.

A few core ideas about hyperbolic functions:

• They include definitions like \( \sinh(x) \), \( \cosh(x) \), \( \tanh(x) \), and importantly, reciprocal forms such as \( \frac{e}{x} \).
• These functions often appear in physics, engineering, and mathematical modeling due to their unique properties, covering sector areas in hyperbolas.
• Hyperbolic curves change direction depending on the quadrants they span, reflecting either vertical or horizontal asymptotic behaviors.

In volume problems, hyperbolic functions serve as crucial components to identify regions of intersection and set up boundaries for integration.