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Magazine Chapter 6: Problem 25
Find the volume of the solid that results when the region enclosed by the given curves is revolved about the \(y\) -axis. $$ y=\ln x, x=0, y=0, y=1 $$
Short Answer
Expert verified
The volume is \( \frac{\pi}{2} (e^2 - 1) \).
Step by step solution
01
Understanding the Problem
We need to find the volume of the solid generated when the region bounded by the curves \( y = \ln x \), \( x = 0 \), \( y = 0 \), and \( y = 1 \) is revolved about the \( y \)-axis. This requires integrating the area of circular disks.
02
Change Variables
Since we're revolving around the \( y \)-axis, we need to express \( x \) in terms of \( y \). From \( y = \ln x \), solve for \( x \) to get \( x = e^y \).
03
Determine Limits of Integration
Identify the limits of \( y \) using the boundaries \( y = 0 \) and \( y = 1 \). Hence, the limits of integration for \( y \) are 0 to 1.
04
Set Up the Integral
The volume \( V \) of the solid of revolution is given by the formula \[ V = \pi \int_{a}^{b} x^2 \: dy \], where \( x = e^y \) and the limits are from \( a = 0 \) to \( b = 1 \). Substitute \( x = e^y \) into the formula to get \[ V = \pi \int_{0}^{1} (e^y)^2 \: dy = \pi \int_{0}^{1} e^{2y} \: dy \].
05
Evaluate the Integral
Calculate the integral \[ \int e^{2y} \: dy \]. Find the antiderivative: \[ \int e^{2y} \: dy = \frac{1}{2}e^{2y} + C \]. Evaluate it from 0 to 1:\[ V = \pi \left[ \frac{1}{2} e^{2y} \right]_{0}^{1} = \pi \left( \frac{1}{2} e^{2 \times 1} - \frac{1}{2} e^{2 \times 0} \right) \].
06
Simplify the Result
Substitute the limits into the antiderivative and simplify the terms:\[ V = \pi \left( \frac{1}{2} e^2 - \frac{1}{2} \times 1 \right) = \frac{\pi}{2} (e^2 - 1) \].
07
Final Result
The volume of the solid of revolution is \[ V = \frac{\pi}{2} (e^2 - 1) \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Calculus
Integral calculus is a fundamental concept in mathematics, mainly used to find areas, volumes, and other quantities under a curve. When dealing with functions, integration allows us to calculate the accumulation of quantities. In our case, it's used to calculate the volume of a solid that forms when a region is revolved around an axis.
- Integration involves finding the integral or antiderivative of a function, which represents the function's accumulated values.
- The definite integral calculates the exact value of this accumulation between two limits, providing a precise area or volume.
- In the current exercise, integral calculus is used to find the volume of the solid by setting up and solving the integral of circular disks along the y-axis.
Solid of Revolution
A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional area around an axis. This geometric concept helps visualize how regions bounded by curves can become tangible shapes.
- The solid's shape depends on the curves and the axis of rotation, presenting unique volume challenges.
- When revolving around the y-axis, as in this exercise, you translate x-values to the equivalent rotation, changing the integration approach.
- Understanding solids of revolution bridges the gap between two-dimensional calculations and real-world volumetric analysis.
Natural Logarithm
The natural logarithm function, denoted by \( \ln \), is the logarithm with base \( e \), where \( e \) is approximately 2.71828. It is a critical function in calculus due to its simplicity and properties.
- \( \ln x \) describes the power to which \( e \) must be raised to achieve the value \( x \).
- When manipulating functions involving \( \ln x \), such as the equation \( y = \ln x \), you solve for \( x \) using exponentiation: \( x = e^y \).
- This process aligns x-values with their counterparts on the curve, vital when applying the disk method.
Disk Method
The disk method is a technique in integral calculus used to find the volume of a solid of revolution. This approach involves visualizing the solid as a stack of disks or thin circular slices.
- Each disk represents a cross-sectional area, with thickness dependent on the variable of integration.
- For revolving around the y-axis, the radius of each disk is derived from the function given for x, such as \( x = e^y \).
- The volume of each disk is \( \pi \times \text{radius}^2 \), and integrating these volumes from the lower to upper limit accumulates the total solid volume.
