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Chapter 6: Problem 4

\(1 - 18\) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. $$ y = \sqrt { 25 - x ^ { 2 } } , y = 0 , x = 2 , x = 4 ; \quad \text { about the } x $$

Short Answer

Expert verified
The volume of the solid is \(\frac{94\pi}{3}\).

Step by step solution

01

Understand the Problem

We need to find the volume of the solid generated by rotating the region bounded by the curves \(y = \sqrt{25 - x^2}\), \(y = 0\), \(x = 2\), and \(x = 4\) around the \(x\)-axis. The first curve is part of a semicircle with radius 5 centered at the origin. The other boundaries are horizontal and vertical lines.
02

Identify the Bounded Region

Sketch the bounded region: \(y = \sqrt{25 - x^2}\) is the upper semicircle of \((x^2 + y^2 = 25)\). The lines \(x=2\) and \(x=4\) are vertical lines that cut through the semicircle, and \(y=0\) is the \(x\)-axis, which forms the bottom boundary. This creates a sector of the semicircle from \(x = 2\) to \(x = 4\).
03

Set Up the Volume Integral

The volume of the solid of revolution can be found using the disk method. At any \(x\), the radius of the disk is \(y = \sqrt{25 - x^2}\). The volume integral is then:\[ V = \pi \int_{2}^{4} (\sqrt{25 - x^2})^2 \, dx = \pi \int_{2}^{4} (25 - x^2) \, dx. \]
04

Simplify the Integral

Simplify the integrand:\[ \pi \int_{2}^{4} (25 - x^2) \, dx = \pi \left( \int_{2}^{4} 25 \, dx - \int_{2}^{4} x^2 \, dx \right). \] This simplifies to two separate integrals.
05

Evaluate the Integrals

Evaluate each integral separately:- \( \int_{2}^{4} 25 \, dx = 25[x]_{2}^{4} = 25(4 - 2) = 50 \).- \( \int_{2}^{4} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{2}^{4} = \frac{64}{3} - \frac{8}{3} = \frac{56}{3} \).The result is:\[ V = \pi \left( 50 - \frac{56}{3} \right) = \pi \times \frac{150 - 56}{3} = \pi \times \frac{94}{3}. \]
06

Compute the Final Volume

Which results in:\[ V = \frac{94\pi}{3}. \]This is the volume of the solid.

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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 powerful tool for finding the volume of a solid of revolution. When a region in the plane is rotated around an axis, the resulting 3D object resembles a series of disks stacked along that axis. To use the Disk Method, think of each disk as having a small thickness, and calculate the volume of each thin disk.
  • Each disk has a radius equal to the distance from the axis of rotation to the curve.
  • The thickness of each disk is a small change along the axis of rotation, denoted as \(dx\).
  • The formula for the volume of each disk is \( V = \pi \, r^2 \, dx \), where \( r \) is the radius and \( dx \) is the thickness.
To find the total volume, integrate the volume of these disks over the interval of the bounding limits. This method is especially useful when the entire area from the curve to the axis is solid.
Definite Integral
A Definite Integral is a critical concept in calculus, especially in finding volumes, areas, and other quantities that involve accumulation. It is represented as an integral with specified upper and lower limits, \(\int_{a}^{b} \). By using definite integrals, you can calculate the accumulated quantity of a function between these limits.
  • The \(a\) and \(b \) in \(\int_{a}^{b} \) are the lower and upper limits of integration.
  • In the context of volume, this calculates the sum of an infinite number of small disks (or washers) from \(a\) to \(b\).
  • The area under the curve, when rotated around an axis, becomes a volume in 3D space.
For the problem given, the definite integral is used to find the volume between \( x = 2 \) and \( x = 4 \), exploiting the disk method for the semicircular region.
Semicircle
The region we're interested in has a semicircular boundary, derived from the equation \( y = \sqrt{25-x^2} \). This equation describes a semicircle with radius 5 on the coordinate plane:
  • Equation \(x^2 + y^2 = 25\) describes a full circle centered at the origin with radius 5.
  • The given function \(y = \sqrt{25-x^2}\) represents just the upper half, hence it's a semicircle.
  • This semicircle sets one of the bounds of the solid of revolution.
The semicircle's curve is crucial as it defines the top boundary of the disks used in the volume calculation. By rotating this curve between \(" x = 2\) and \(x = 4\), we can use it to determine volumes accurately.
Bounded Region
Before you can calculate the volume of a solid of revolution, you must identify the Bounded Region of interest. This is the area enclosed by the curves and lines specified in the problem.
  • The semicircle \(y = \sqrt{25-x^2}\) forms the top boundary.
  • Vertical lines \(x=2\) and \(x=4\) serve as the side boundaries.
  • The line \(y=0\) (the x-axis) establishes the lower boundary.
In this exercise, these boundaries create a pie-shaped slice from the semicircle between \(x = 2\) and \(x = 4\). Understanding this bounded region is essential because it dictates where you will integrate to find the volume.

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Most popular questions from this chapter

Use Newton's Law of Gravitation to compute the work required to launch a 1000 -kg satellite vertically to an orbit 1000 \(\mathrm{km}\) high. You may assume that the earth's mass is \(5.98 \times 10^{24} \mathrm{kg}\) and is concentrated at its center. Take the radius of the earth to be \(6.37 \times 10^{6} \mathrm{m}\) and \(G=6.67 \times 10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}.\)

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