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Chapter 6: Problem 5

Sketch the region \(R\) bounded by the graphs of the equations, and find the volume of the solid generated if \(R\) is revolved about the indicated axis. \(y=\sqrt{x}, \quad x=4, \quad y=0 ; \quad y\) -axis

Short Answer

Expert verified
The volume is \(\frac{32\pi}{5}\).

Step by step solution

01

Graph the curves

Graph the function \(y = \sqrt{x}\) from \(x = 0\) to \(x = 4\); this is a curve starting at the origin (0,0) and ending at (4,2). Draw a line at \(x = 4\) which is a vertical line, and the horizontal line \(y = 0\), which is the x-axis. The region bounded by these curves is the area we are interested in.
02

Understand the region for revolution

The region \(R\) is bounded by the curve \(y = \sqrt{x}\), the line \(x = 4\), and the x-axis \(y = 0\), and we are revolving this region around the y-axis.
03

Set up the integral for volume using the disk method

We will use the disk method to find the volume. As we revolve around the y-axis, the thickness of each disk is \(\Delta y\) and its radius is \(x = y^2\). The volume of each disk is \(\pi (radius)^2 (thickness) = \pi (y^2)^2 \Delta y = \pi y^4 \Delta y\).
04

Determine the limits of integration

Since the region starts at \(x = 0\) which corresponds to \(y = 0\) and ends at \(x = 4\) which corresponds to \(y = 2\), the limits for \(y\) are from 0 to 2.
05

Integrate to find the volume

Set up the integral: \[V = \int_{0}^{2} \pi y^4 \, dy = \pi \int_{0}^{2} y^4 \, dy.\]Evaluate the integral: \[V = \pi \left[ \frac{y^5}{5} \right]_{0}^{2} = \pi \left( \frac{32}{5} - 0 \right) = \frac{32 \pi}{5}.\]
06

Conclusion

The volume of the solid generated when the region \(R\) is revolved about the y-axis is \(\frac{32 \pi}{5}\).

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Key Concepts

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

Disk Method
The disk method is a valuable technique for finding the volume of a solid of revolution. Imagine slicing your solid into tiny disks, much like slicing a loaf of bread. Each disk has a small thickness and a circular face. By summing up the volumes of each disk from one point to another, you can find the volume of the entire solid.

When you revolve a region around an axis, each disk's face is perpendicular to the axis of revolution. In this exercise, we revolve around the y-axis, meaning our disks are horizontal. The thickness of each disk is expressed as \(\Delta y\) since we are moving along the y-axis.
  • Radius of each disk: The radius is determined by the x-value of the function, which is expressed as \(x = y^2\).
  • Volume of each disk: The volume is calculated using the formula \( \pi \times (\text{radius})^2 \times (\text{thickness}) = \pi y^4 \Delta y \).
By integrating these small volumes, the total volume of the solid is found.
Definite Integral
The definite integral is an essential tool in calculus used to calculate areas under curves, among other things, including volumes of revolution. It can be thought of as an infinite summation of infinitesimally small areas. In the context of finding volumes, here it aids in summing up the volumes of individual disks to get the total volume.

For the problem, we set up an integral to find the volume of the solid generated by revolving the region around the y-axis. The integral expression reflects the continuous addition of disk volumes from the lowest point \(y = 0\) to the highest point \(y = 2\).
  • Setting up the integral: The integral we use is \( \int_{0}^{2} \pi y^4 \, dy \).
  • Evaluate the integral: Calculating this yields: \(\pi \left[ \frac{y^5}{5} \right]_{0}^{2} = \frac{32 \pi}{5} \).
This result represents the volume of the solid.
Solid of Revolution
A solid of revolution is formed when a two-dimensional shape is rotated around a line (axis). This creates a three-dimensional object that has rotational symmetry. In this exercise, by revolving the bounded region around the y-axis, a solid structure is formed.

Understanding how these solids are generated helps in visualizing and solving such problems efficiently. Here's a basic breakdown:
  • Axis of revolution: This is the line around which the region is rotated. In our scenario, it's the y-axis.
  • Bounded region: Here, it's defined by \(y = \sqrt{x}\), \(x = 4\), and \(y = 0\). These boundaries hold the possible area that, when revolved, forms the solid.
  • Resulting solid: After performing the revolution, a symmetrical, typically round solid is obtained, resembling a vase or pod depending on the boundaries.
Visualizing this can sometimes be challenging, but drawing sketches helps concretize the concept.

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Most popular questions from this chapter

Sketch the region bounded by the graphs of the equations, and find \(m, M_{x}, M_{x},\) and the centroid. $$ y^{2}=x, \quad 2 y=x $$

Sketch the region \(R\) bounded by the graphs of the equations, and find the volume of the solid generated by revolving \(R\) about the indicated axis. \(y=x^{4}, \quad y=0, \quad x=1 ; \quad y\) -axis

Find the arc length of the graph of \(y=\frac{3 x^{8}+5}{30 x^{3}}\) from \(\left(1, \frac{4}{5}\right)\) to \(\left(2, \frac{773}{240}\right) .\)

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The population of a city has increased since 1985 at a rate of \(1.5+0.3 \sqrt{t}+0.006 t^{2}\) thousand people per year, where \(t\) is the number of years after 1985 . Assuming that this rate continues and that the population was 50,000 in \(1985,\) estimate the population in \(1994 .\)
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