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Magazine Chapter 8: Problem 10
The region is rotated around the x-axis. Find the volume. $$\text { Bounded by } y=\cos x, y=0, x=0, x=\pi / 2$$
Short Answer
Expert verified
Volume = \( \frac{\pi^2}{4} \).
Step by step solution
01
Understand the Problem
We need to find the volume of the solid formed when the region bounded by the curve \( y = \cos x \), the lines \( y = 0 \), \( x = 0 \), and \( x = \pi/2 \) is rotated around the x-axis.
02
Set Up the Integral
The formula for the volume of a solid of revolution around the x-axis is \[V = \pi \int_{a}^{b} [f(x)]^2 \, dx\]Here, \( f(x) = \cos x \), \( a = 0 \), and \( b = \pi/2 \). Thus, the integral becomes: \[V = \pi \int_{0}^{\pi/2} (\cos x)^2 \, dx\]
03
Simplify the Integral
Use the trigonometric identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) to simplify the integral: \[V = \pi \int_{0}^{\pi/2} \frac{1 + \cos(2x)}{2} \, dx\]This simplifies to: \[V = \frac{\pi}{2} \int_{0}^{\pi/2} (1 + \cos(2x)) \, dx\]
04
Evaluate the Integral
Now, evaluate the integral:\[\frac{\pi}{2} \left[ \int_{0}^{\pi/2} 1 \, dx + \int_{0}^{\pi/2} \cos(2x) \, dx \right]\]The first integral, \( \int_{0}^{\pi/2} 1 \, dx \), is \( x \bigg|_{0}^{\pi/2} \), which evaluates to \(\pi/2\).The second integral, \( \int_{0}^{\pi/2} \cos(2x) \, dx \), can be solved using a substitution: let \( u = 2x \), then \( du = 2 \, dx \) or \( dx = du/2 \). This transforms the limits from \( 0 \) to \( \pi \), making the integral \(\frac{1}{2} \int_{0}^{\pi} \cos(u) \, du\), which evaluates to zero (since the integral of one period of cosine is zero).Thus:\[V = \frac{\pi}{2} \left[ \pi/2 + 0 \right] = \frac{\pi^2}{4}.\]
05
Conclude the Result
The calculated volume of the solid formed by rotating the given region around the x-axis is \( \frac{\pi^2}{4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
The concept of definite integrals is a foundational element in calculus that helps us calculate the accumulation of quantities, like area or volume, over a certain interval. For the problem at hand, we're concerned with finding the volume of a solid of revolution by integrating over a defined range. In this case, the integral is taken from the lower bound, \( x = 0 \), to the upper bound, \( x = \frac{\pi}{2} \).
A definite integral not only helps find areas but is also key in determining volumes by revolving a region around an axis. By computing \( \pi \int_{0}^{\pi/2} [\cos(x)]^2 \, dx \), we ascertain the volume of the solid formed. This integral uses the limits \( 0 \) to \( \pi/2 \), matching the specified intervals for the function \( y = \cos x \).
The definite integral transforms abstract mathematical reasoning into a concrete numerical result, facilitating the calculation of real-world quantities like volume, area, and even length.
A definite integral not only helps find areas but is also key in determining volumes by revolving a region around an axis. By computing \( \pi \int_{0}^{\pi/2} [\cos(x)]^2 \, dx \), we ascertain the volume of the solid formed. This integral uses the limits \( 0 \) to \( \pi/2 \), matching the specified intervals for the function \( y = \cos x \).
The definite integral transforms abstract mathematical reasoning into a concrete numerical result, facilitating the calculation of real-world quantities like volume, area, and even length.
Trigonometric Identities
Trigonometric identities are handy tools to simplify and solve calculus problems. In solving for the volume of a solid of revolution in this exercise, the trigonometric identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \) proves invaluable. Why?
This identity allows us to express \( (\cos x)^2 \) in a form that is easier to integrate. By converting \( (\cos x)^2 \) to \( \frac{1 + \cos(2x)}{2} \), a complex trigonometric expression is turned into a combination of constants and a double-angle cosine function.
This identity allows us to express \( (\cos x)^2 \) in a form that is easier to integrate. By converting \( (\cos x)^2 \) to \( \frac{1 + \cos(2x)}{2} \), a complex trigonometric expression is turned into a combination of constants and a double-angle cosine function.
- This simplification is critical when integrating, as direct integration of \( (\cos x)^2 \) is not straightforward.
- It avoids more complicated integration techniques and paves the way for using simpler integrals such as \( \int \cos(kx) \, dx \).
Calculus Problem Solving
The art of calculus problem-solving involves breaking down problems into understandable steps, using calculus principles like differentiation and integration. When solving this type of problem, one must:
- **Understand the problem:** Clearly analyze what the problem is asking for, such as the volume of a rotated solid.
- **Set up the problem:** Determine the integral needed to calculate the volume or area.
Ultimately, applying these steps systematically ensures problems are less daunting and solutions are more achievable.
- **Understand the problem:** Clearly analyze what the problem is asking for, such as the volume of a rotated solid.
- **Set up the problem:** Determine the integral needed to calculate the volume or area.
- Identify the function or functions that define the region.
- Determine the appropriate limits of integration.
- Incorporate simplification strategies, like trigonometric identities, to make integration feasible.
- Use substitution or integration by parts when necessary.
Ultimately, applying these steps systematically ensures problems are less daunting and solutions are more achievable.
