Question

Find the volume of the solid generated by revolving the region bounded by the curve $y=4\cos x$ and the $x$ -axis, $\frac{\pi }{2}\le x\le \frac{3\pi }{2}$, about the $x$ -axis. (Express the answer in exact form.)

Short answer

The volume of the solid is $8{\pi }^2$ cubic units.

Step-by-step solution

Understand the Problem

We need to find the volume of a solid of revolution formed by rotating the region bounded by $y=4\cos x$ and the $x$-axis from $x=\frac{\pi }{2}$ to $x=\frac{3\pi }{2}$ around the $x$-axis. This region forms a solid shape when revolved around the $x$-axis.

Set Up the Integral

To find the volume of the solid of revolution, we use the formula for the volume of a solid of revolution around the $x$-axis: $$V=\pi {\int }_a^b[f(x){]}^{2}dx$$Here, $f(x)=4\cos x$, and the bounds $a$ and $b$ are $\frac{\pi }{2}$ and $\frac{3\pi }{2}$, respectively. So, $$V=\pi {\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}(4\cos x{)}^{2}dx$$

Simplify the Integral

Simplify the expression inside the integral:$$(4\cos x{)}^2=16{\cos }^{2}x$$Substituting back, the volume integral becomes:$$V=\pi {\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}16{\cos }^{2}xdx$$Factor out the constant:$$V=16\pi {\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}{\cos }^{2}xdx$$

Solve the Integral Using Trigonometric Identity

To integrate ${\cos }^{2}x$, use the identity:$${\cos }^{2}x=\frac{1+\cos (2x)}{2}$$Substitute this into the integral:$$V=16\pi {\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}\frac{1+\cos (2x)}{2}dx$$This becomes two separate integrals:$$V=16\pi ({\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}\frac{1}{2}dx+{\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}\frac{\cos (2x)}{2}dx)$$

Evaluate the Integrals

The first integral evaluates to:$${\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}\frac{1}{2}dx=\frac{1}{2}[x{]}_{\frac{\pi }{2}}^{\frac{3\pi }{2}}=\frac{1}{2}(\frac{3\pi }{2}-\frac{\pi }{2})=\frac{1}{2}\pi$$The second integral:$${\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}\frac{\cos (2x)}{2}dx=\frac{1}{2}{\left[\frac{\sin (2x)}{2}\right]}_{\frac{\pi }{2}}^{\frac{3\pi }{2}}$$Evaluating this from $\frac{\pi }{2}$ to $\frac{3\pi }{2}$, we get:$$\frac{1}{4}[\sin (3\pi )-\sin (\pi )]=\frac{1}{4}(0-0)=0$$

Combine Results and Calculate Volume

Add the results from the integrals:$$V=16\pi (\frac{1}{2}\pi +0)=16\pi \left(\frac{1}{2}\pi \right)=8{\pi }^2$$

State the Final Answer

The volume of the solid generated by revolving the region around the $x$-axis is $8{\pi }^2$ cubic units.

Key concepts

Definite Integrals

In this problem, we use definite integrals to find the volume of a solid. A definite integral helps us calculate the exact area under a curve between two specific limits. Here, the curve is given by the function $y=4\cos x$, and the limits are from $\frac{\pi }{2}$ to $\frac{3\pi }{2}$. To find the volume, we revolve the area under this curve about the $x$-axis.
This approach uses the formula for the volume of a solid of revolution:

• $$V=\pi {\int }_a^b[f(x){]}^{2}dx$$

By substituting $f(x)=4\cos x$ and the limits, we set up a definite integral which allows us to compute the required volume accurately. Definite integrals are powerful as they provide exact areas and volumes without the need for approximation.

Trigonometric Identities

To solve the integral from our exercise, we need to simplify ${\cos }^{2}x$ using trigonometric identities. An important identity here is the double-angle formula:

• $${\cos }^{2}x=\frac{1+\cos (2x)}{2}$$

This identity transforms our integral into a simpler form. Trigonometric identities like this one are mathematical tools that allow us to rewrite expressions in alternative forms to make them easier to integrate or differentiate.
In this problem, applying this identity helps break down the integral into two simpler parts, making it more feasible to solve. Knowing and using these identities is essential when dealing with trigonometric functions in calculus.

Integration Techniques

Solving integrals often requires specific techniques, and in this problem, we encounter several. Once we've simplified the function using trigonometric identities, we integrate each term separately.
Integration can involve different methods, such as:

• Basic antiderivatives, which is needed for terms like $\int \frac{1}{2}dx$.
• Substitution, particularly beneficial when dealing with more complex functions.

For the term $\frac{\cos (2x)}{2}$, substitution or recognition of standard antiderivatives helps us perform the calculation:

• $$\int \cos (2x)dx=\frac{1}{2}\sin (2x)+C$$

Understanding various integration techniques is crucial. It equips students to handle different problems by knowing the right approach to take for each unique integral.