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Chapter 6: Problem 21

\(21-26\) Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $$ y=\ln x, y=0, x=2 ; \quad \text { about the } y-axis $$

Short Answer

Expert verified
The integral setup is \( V = 2\pi \int_{1}^{2} x \, \ln x \, dx \).

Step by step solution

01

Identify the curves and boundaries

The boundaries given are the curve \( y = \ln x \), the line \( y = 0 \), and \( x = 2 \). These define a region that needs to be rotated about the y-axis.
02

Decide on the method to use

Since the region is being rotated around the y-axis, we will use the method of cylindrical shells to find the volume.
03

Setup the integral using cylindrical shells

The formula for volume using cylindrical shells method is \[ V = 2\pi \, \int_{a}^{b} x \, (f(x) - g(x)) \, dx \] where \( x \) is the radius to the y-axis, and \( f(x) - g(x) \) is the height of the shell. Here, the integrand will be \( x(\ln x - 0) \), the height of the shell is \( \ln x \), and the bounds are from 1 (since \( y=0 \) gives \( x=1 \)) to 2.
04

Write the integral expression

The integral expression to define the volume is: \[ V = 2\pi \, \int_{1}^{2} x \, \ln x \, dx \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylindrical Shells Method
The cylindrical shells method is an elegant technique used to find volumes of solids of revolution. Specifically, it is helpful when rotating a region around a vertical line, such as the y-axis. By visualizing this method, imagine unrolling a label from a can. Each label represents a cylindrical shell, and stacking these shells constructs the entire solid. This strategy is powerful for symmetry and efficiency in solving these volume problems.

To apply this method, consider these essential steps:
  • Identify the radius of the shell, which is the distance from the axis of rotation to the shell. In this example, it's the x-coordinate since the rotation is around the y-axis.
  • Determine the height of the shell, which is defined by the function's value in the region, such as \( f(x) - g(x) \) where \( f(x) \) exceeds \( g(x) \) over an interval.
  • Calculate the integral, incorporating the formula for cylindrical shells: \[ V = 2\pi \, \int_{a}^{b} x \, (f(x) - g(x)) \, dx \]
Using these steps aligns with visualizing the entire shell structure and unraveling a complex volume problem into a manageable integral.
Integrals in Calculus
Integrals in calculus are a fundamental concept used to calculate areas, volumes, and even to solve differential equations. Understanding integrals deeply aids in solving problems involving accumulated quantities, such as under a curve or within a solid of revolution. When using the method of cylindrical shells, integrals help define the precise volume of these shells.

In the provided solution, we set up an integral that reflects the volume of the solid. Here鈥檚 why and how:
  • The integral reflects summing up infinitesimally small elements - in this case, the cylindrical shells that form a solid shape.
  • It defines the bounds \( a \) and \( b \), which denote where to start and stop accumulating the shells. For the example, the bounds are from 1 to 2.
  • This approach captures both the geometric interpretation and the precise mathematical mechanism, where the integral \( \int_{1}^{2} x \, \ln x \, dx \) is set up but not solved.
Integrals are not always straightforward, but breaking them into meaningful components can provide clarity and simplicity in handling complex calculus problems.
Rotation About the Y-Axis
Rotation about the y-axis is a common scenario in calculus problems involving solids of revolution. By twisting a 2D region around this vertical axis, a 3D object is formed, offering intriguing geometrical properties. It becomes vital to visualize this scenario to understand how initial shapes translate into volume.

Key points to consider:
  • Identify the region to be rotated, recognizing the boundaries initially provided by the functions and lines.
  • Determine how the area between these boundaries spins to form a deductive 3D shape.
  • Utilize methods like cylindrical shells to simplify the problem. This approach allows translating a curved shape into cylindrical segments for practical integration.
When dealing with shapes formed by rotation around the y-axis, recognizing the axis, and symmetry in the problem is pivotal. This recognition simplifies setup steps and clarifies integral formation, ultimately accelerating problem-solving with precision and efficiency.

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Most popular questions from this chapter

\(15-20\) Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell. $$y=\sqrt{x}, y=0, x=1 ; \quad \text { about } x=-1$$

31-34 Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. $$ \begin{array} { l } { y = e ^ { - x ^ { 2 } } , y = 0 , x = - 1 , x = 1 } \\\ { \text { (a) About the } x \text { -axis } } \end{array} \text { (b) About } y = - 1 $$

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (ln fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas. (a) A barrel with height \(h\) and maximum radius \(R\) is con- structed by rotating about the \(x\) -axis the parabola \(y = R - c x ^ { 2 } , - h / 2 \Rightarrow x \leq h / 2 ,\) where \(c\) is a positive constant. Show that the radius of each end of the barrel is \(r = R - d ,\) where \(d = c h ^ { 2 } / 4\) (b) Show that the volume enclosed by the barrel is $$V = \frac { 1 } { 3 } \pi h \left( 2 R ^ { 2 } + r ^ { 2 } - \frac { 2 } { 5 } d ^ { 2 } \right)$$

\(5-28\) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y .\) Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $$y=3 x^{2}, \quad y=8 x^{2}, \quad 4 x+y=4, \quad x \geqslant 0$$

A force of 10 \(\mathrm{lb}\) is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?
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