Question

IThe region bounded between the curve \(y=\frac{1}{\sqrt{1+\cos x}}, 0.3 \leq x \leq 1.1\), and the \(x\) -axis is revolved about the \(x\) -axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)

Short answer

The volume is approximately 1.31.

Step-by-step solution

Set Up the Formula for Volume

The volume of a solid formed by rotating a bounded region around the x-axis is given by the formula \( V = \pi \int_{a}^{b} [f(x)]^2 \,dx \), where \( f(x) \) is the function of the curve being revolved. Here, \( f(x) = \frac{1}{\sqrt{1+\cos x}} \), \( a = 0.3 \), and \( b = 1.1 \).

Square the Function

Square the function \( f(x) \): \[ [f(x)]^2 = \left( \frac{1}{\sqrt{1+\cos x}} \right)^2 = \frac{1}{1+\cos x} \].

Substitute and Simplify the Integral

Substitute \( [f(x)]^2 \) into the volume formula: \[ V = \pi \int_{0.3}^{1.1} \frac{1}{1+\cos x} \, dx \]. This integral can be simplified using a trigonometric identity: \[ \frac{1}{1+\cos x} = \frac{1}{2} \sec^2 \left( \frac{x}{2} \right) \].

Use the Identity to Transform the Integral

Using the identity from Step 3, the integral becomes: \[ V = \frac{\pi}{2} \int_{0.3}^{1.1} \sec^2\left( \frac{x}{2} \right) \, dx \].

Integrate Using a Table of Integrals

Find the antiderivative of \( \sec^2 \left( \frac{x}{2} \right) \): \[ \int \sec^2(\frac{x}{2}) \ dx = 2 \tan\left(\frac{x}{2}\right) + C \]. Now apply the definite integral: \[ V = \frac{\pi}{2} [2 \tan\left(\frac{x}{2}\right)]\bigg|_{0.3}^{1.1} \].

Evaluate the Definite Integral

Compute the definite integral: \[ V = \pi \left( \tan\left(\frac{1.1}{2}\right) - \tan\left(\frac{0.3}{2}\right) \right) \]. Calculate the values to get the volume: \[ V \approx \pi \times (0.569 - 0.151) \approx \pi \times 0.418 \approx 1.31 \].

Round the Result

Round the result to two decimal places: The final volume of the solid is approximately \( 1.31 \).

Key concepts

Definite Integrals

Definite integrals are a powerful tool in calculus used to calculate the accumulation of quantities, such as area or volume. A definite integral is represented as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration. In the context of this exercise, we use the definite integral to calculate the volume of a solid by revolving a region between the curve and the x-axis.
To solve a definite integral, you need to find an antiderivative of the function \(f(x)\), then evaluate it at the upper and lower limits, \(b\) and \(a\). The result of the integral provides the exact amount of the quantity of interest within the given bounds, in this case, volume.
By treating the problem as a revolution of a curve, the definite integral helps in determining the total volume enclosed by the solid, ensuring an accurate calculation.

Volume of Solids of Revolution

The volume of solids of revolution is calculated using a specific integral formula. Solids of revolution occur when a region is rotated around a line, such as the x-axis or y-axis, to create a three-dimensional object. The formula to determine the volume is \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \), where \(f(x)\) is the function describing the curve being rotated.
This formula is derived from the concept of cross-sectional discs or washers, whose radii are defined by the function \(f(x)\). Each infinitesimal disc has a small volume, \(dV\), and these aggregate to give the total volume of the solid.
In our exercise, the function \(f(x) = \frac{1}{\sqrt{1+\cos x}}\) is squared since it defines the radius of revolution. The limits of integration \(0.3\) and \(1.1\) reflect the interval over which the region is revolved, ensuring the correct calculation of the total volume.

Trigonometric Identities

Trigonometric identities are essential tools in calculus, particularly when simplifying integrals involving trigonometric functions. These identities can transform complex expressions into simpler, more manageable forms.
In the given problem, we encountered the expression \( \frac{1}{1+\cos x} \). By utilizing a trigonometric identity, it was rewritten as \( \frac{1}{2} \sec^2\left( \frac{x}{2} \right) \). This simplification allowed for easier integration.
The identification and application of such identities often involve using well-known formulas, such as the Pythagorean identities or the angle sum and difference formulas. Learning to recognize and apply these identities can significantly streamline the process of solving integrals in calculus.

Table of Integrals

A Table of Integrals is a valuable resource for finding antiderivatives when faced with complex or non-standard integral forms. It provides a collection of integral formulas that can be directly applied, saving time and effort in deriving the antiderivative from scratch.
In this exercise, we used the Table of Integrals to identify the antiderivative of \( \sec^2(\frac{x}{2}) \), knowing that \( \int \sec^2(x) \, dx = \tan(x) \). The table helped translate this to \( \int \sec^2(\frac{x}{2}) \, dx = 2 \tan\left(\frac{x}{2}\right) + C \), which was then used to evaluate the integral over the specified interval.
Using such a table is a practical approach in calculus, especially for integrals that are too complex to solve by basic antiderivative techniques, enabling efficient problem-solving and understanding complex calculus concepts.