/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Let \(R\) be the region bounded ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 37

Let \(R\) be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=x^{3}, y=0, x=2$$

Short Answer

Expert verified
Question: Determine the volume of the solid generated when the region R bounded by the curves \(y=x^3\), \(y=0\) (the x-axis), and \(x=2\) is revolved around the \(y\)-axis using the disk or washer method. Answer: The volume of the solid generated when R is revolved about the y-axis is \(32\pi\) cubic units.

Step by step solution

01

Sketch the region R

Before diving into the problem, start by sketching the region R to gain a better understanding of the shape being revolved. The region is bounded by \(y=x^3\), \(y=0\), and \(x=2\). The x-axis bounds R from below, so there is no "hole" in the solid when revolved.
02

Identify the Disk Thickness, Outer Radius, and Inner Radius

Rotate R about the \(y\)-axis. Since we're using the disk/washer method, take thin disks perpendicular to the axis of rotation (the y-axis) with thickness \(\Delta y\). The outer radius of each disk is \(x = 2\). Since R is bounded by \(y=0\) on the bottom, there is no hole in the resulting solid, and thus the inner radius is 0.
03

Set up the Volume Integral

Using the disk method, the volume of a single disk is given by \(\pi(\text{outer radius}^2 - (\text{inner radius})^2)\Delta y\). To find the volume of the entire solid, integrate the volume of each disk along the y-axis. In this case, \(x=2\) is the outer boundary, and \(y=0\) at the \(x\)-axis is the inner boundary. The integral takes the following form: $$V = \pi \int_{y=a}^{y=b} [(2)^2 - (0)^2]dy$$
04

Determine the Limits of Integration

To find the limits of integration, we need to express \(y\) in terms of \(x\). We know that \(y=x^3\), \(x=0\), and \(x=2\). With k boundaries are already defined in terms of y coordinates for us: 1. When \(x=0\), \(y=0^3=0\) 2. When \(x=2\), \(y=2^3=8\) Thus, the limits of integration are \(y=0\) to \(y=8\). The integral becomes: $$V = \pi \int_{0}^{8} [(2)^2 - (0)^2]dy$$
05

Evaluate the Integral and Compute the Volume

Now, we can evaluate the integral. The volume V is constant with respect to y, so it can be factored out of the integral. $$V = \pi [(2)^2 - (0)^2] \int_{0}^{8} dy$$ After the integration, we find: $$V = \pi [(2)^2 - (0)^2] [y|_{0}^{8}] = \pi [(2)^2 - (0)^2] [8 - 0]$$ Finally, compute the volume: $$V = \pi (4)(8) = 32\pi$$ The volume of the solid generated when R is revolved about the y-axis is \(32\pi\) cubic units.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Calculus
Calculus is a branch of mathematics focused on the study of change and motion. Through its two major sub-fields, differential calculus and integral calculus, it gives us the tools to analyze and solve problems involving curves, areas, volumes, and rates of change.

In the context of the disk method volume calculation problem, calculus comes into play when we need to determine the volume of a solid of revolution. Using calculus, specifically integral calculus, we can sum an infinite number of infinitesimally thin disks to find the total volume of the solid. This process of summation is what integration is all about: adding up pieces to find a whole.
Solid of Revolution
A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional shape around an axis. The resulting solid has a symmetry around the axis of rotation, similar to objects like vases or bottles.

The exercise provided involves creating a solid of revolution by revolving the region bounded by the curves around the y-axis. The shape we get is akin to a cylinder but with one end possibly being larger or differently shaped than the other. The disk method is particularly useful for calculating volumes of solids of revolution when the revolving axis is either the x-axis or the y-axis, as in this example.
Volume Integral
The volume integral is a way to calculate the volume of a three-dimensional object using integration. For solids of revolution, we often use the disk or washer methods, which involve slicing the solid into thin pieces, computing the volume of each slice, and then summing these volumes via integration.

In our exercise, the volume integral of the solid is simplified because the inner radius is zero, and the outer radius is a constant 2, leading to an integral that is straightforward to evaluate. The integral quantifies the accumulation of all the disk volumes from the bottom to the top of the solid, providing us with the total volume.
Limits of Integration
The limits of integration in an integral denote the boundaries over which the integration is performed. These are crucial for evaluating the definite integral, as they specify the starting and ending points.

For volume calculations, these limits correspond to the bottom and top of the solid of revolution along the axis of rotation鈥攖he y-axis in our example. Setting these limits accurately is essential for obtaining the correct volume. In the given problem, the region R's lowest and highest points on the y-axis are 0 and 8, respectively, providing us with the boundaries needed to solve the integral and determine the solid's volume.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A large tank has a plastic window on one wall that is designed to withstand a force of 90,000 N. The square window is \(2 \mathrm{m}\) on a side, and its lower edge is \(1 \mathrm{m}\) from the bottom of the tank. a. If the tank is filled to a depth of \(4 \mathrm{m},\) will the window withstand the resulting force? b. What is the maximum depth to which the tank can be filled without the window failing?

Find the critical points of the function \(f(x)=\sinh ^{2} x \cosh x.\)

Suppose the acceleration of an object moving along a line is given by \(a(t)=-k v(t),\) where \(k\) is a positive constant and \(v\) is the object's velocity. Assume that the initial velocity and position are given by \(v(0)=10\) and \(s(0)=0,\) respectively. a. Use \(a(t)=v^{\prime}(t)\) to find the velocity of the object as a function of time. b. Use \(v(t)=s^{\prime}(t)\) to find the position of the object as a function of time. c. Use the fact that \(d v / d t=(d v / d s)(d s / d t)\) (by the Chain Rule) to find the velocity as a function of position.

Starting at the same time and place, Abe and Bob race, running at velocities \(u(t)=4 /(t+1) \mathrm{mi} / \mathrm{hr}\) and \(v(t)=4 e^{-t / 2} \mathrm{mi} / \mathrm{hr},\) respectively, for \(t \geq 0\) a. Who is ahead after \(t=5\) hr? After \(t=10\) hr? b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?

Two points \(P\) and \(Q\) are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment \(P Q\) is less than one-fourth the area of the square? Begin by showing that \(x\) and \(y\) must satisfy \(x y<\frac{1}{2}\) in order for the area condition to be met. Then argue that the required probability is \(\frac{1}{2}+\int_{1 / 2}^{1} \frac{d x}{2 x}\) and evaluate the integral.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.