Problem 29 - Calculus Early Transcendentals

Question

Let \(R\) be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=e^{x / 2}, y=e^{-x / 2}, x=\ln 2, x=\ln 3$$

Step-by-Step Solution

Answer

Answer: The volume of the solid generated is \(\frac{1}{2}\pi\) cubic units.

Step 1: Identify the outer and inner radius

Since we are revolving the region around the x-axis, the outer radius in our problem is the distance from the x-axis to the curve \(y=e^{x/2}\), which would simply be the equation itself, so the outer radius is \(r_o(x)=e^{x/2}\). Meanwhile, the inner radius is the distance from the x-axis to the curve \(y=e^{-x/2}\), so the inner radius is \(r_i(x)=e^{-x/2}\).

Step 2: Set up the integral

According to the washer method, the volume of the solid is given by the integral of the difference of the squared outer and inner radii, multiplied by the area of the washer at each x. So, we have the following formula for the volume: $$V = \pi\int_a^b [(r_o(x))^2-(r_i(x))^2]dx$$ In our case, we'll be integrating from \(x=\ln{2}\) to \(x=\ln{3}\): $$V = \pi\int_{\ln{2}}^{\ln{3}} [(e^{x/2})^2-(e^{-x/2})^2]dx$$

Step 3: Simplify the integral

Simplify the expressions inside the integral: $$V = \pi\int_{\ln{2}}^{\ln{3}} [e^x-e^{-x}]dx$$

Step 4: Integrate

Now we integrate the expression with respect to x: $$V = \pi [e^x+e^{-x}]|_{\ln{2}}^{\ln{3}}$$

Step 5: Evaluate the definite integral

We now plug in the limits of integration to obtain the volume: $$V = \pi [(e^{\ln{3}}+e^{-\ln{3}})-(e^{\ln{2}}+e^{-\ln{2}})]$$ Recall that \(e^{\ln{a}}=a\): $$V = \pi [(3+1/3)-(2+1/2)]$$ $$V = \pi [3\frac{1}{3}-2\frac{1}{2}]$$

Step 6: Simplify the expression for the volume

Calculate the result to find the volume of the solid: $$V = \pi [3-\frac{5}{2}]$$ $$V = \frac{1}{2}\pi$$ The volume of the solid generated when the region R is revolved around the x-axis is \(\frac{1}{2}\pi\) cubic units.

Key Concepts

Solid of Revolution

When a two-dimensional shape is rotated around an axis, the result is a three-dimensional object known as a solid of revolution. This method is a key concept in calculus to determine the volume of an object whose cross-section is known. Imagine taking a piece of paper with a shaded region and spinning it around a pencil; the shape you create is analogous to the solid of revolution.

Calculus offers a systematic approach to calculate the volume of such objects using the washer or disk method. These methods involve slicing the solid into thin pieces, calculating the volume of each slice, and summing them up—an application of Riemann sums. The washer method specifically accounts for hollow centers, like a donut shape, hence the name 'washer'.

Definite Integral

The definite integral is crucial for finding the exact volume of solids of revolution. It's a fundamental concept in calculus that quantifies the accumulation of quantities, such as areas under curves and, as in our scenario, volumes. When applied, the definite integral sums up an infinite number of infinitesimally thin washers to find the solid's exact volume.

The process of evaluation involves finding the antiderivative of the function, then applying the Fundamental Theorem of Calculus which suggests subtracting the antiderivative evaluated at the lower limit from that evaluated at the upper limit. The step-by-step solution shows how this principle is applied to find the volume of the solid.

Exponential Functions

Exponential functions play a central role in modeling growth or decay processes, and they have distinctive characteristics such as a constant ratio over equal intervals. In our exercise, we deal with exponential functions of the form y = e^{x/2} and y = e^{-x/2}, where e is the base of the natural logarithm, approximately equal to 2.71828.

These functions grow or decay at a rate proportional to their current value—a critical feature that often occurs in real-world applications like population growth, radioactive decay, or continuously compounded interest. Understanding their behavior is essential when setting up integrals for calculating volumes of solids of revolution since the radius of the washers often depends on these exponential expressions.