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Magazine Chapter 8: Problem 101
Find the area of the surface generated by revolving the curve \(y=\sqrt{x^{2}+2}, 0 \leq x \leq \sqrt{2},\) about the \(x\) -axis.
Short Answer
Expert verified
The surface area is \(4\pi\sqrt{3}\).
Step by step solution
01
Understand the Surface Area Formula for Revolution
When a curve is revolved around the x-axis, the formula for the surface area \( S \) is given by \( S = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \), where \( y \) is the function related to \( x \), and \( \frac{dy}{dx} \) is the derivative of \( y \) with respect to \( x \). This will help in evaluating the surface area for the given range.
02
Differentiate the Given Function
First, find the derivative of \( y = \sqrt{x^2 + 2} \). Using the chain rule, we have \( \frac{dy}{dx} = \frac{d}{dx}(x^2 + 2)^{1/2} = \frac{x}{\sqrt{x^2 + 2}} \).
03
Compute the Expression Inside the Square Root
Substitute \( \frac{dy}{dx} \) into the expression \( 1 + \left( \frac{dy}{dx} \right)^2 \). We have \( 1 + \left( \frac{x}{\sqrt{x^2 + 2}} \right)^2 = 1 + \frac{x^2}{x^2 + 2} = \frac{x^2 + 2 + x^2}{x^2 + 2} = \frac{2x^2 + 2}{x^2 + 2} \). Simplify: \( \frac{2(x^2 + 1)}{x^2 + 2} \).
04
Set Up the Integral for Surface Area
Replace in the surface area formula: \( S = 2\pi \int_{0}^{\sqrt{2}} \sqrt{x^2 + 2} \sqrt{\frac{2(x^2 + 1)}{x^2 + 2}} \, dx \). Simplify under the square root: \( S = 2\pi \int_{0}^{\sqrt{2}} \sqrt{2(x^2 + 1)} \, dx \).
05
Simplify the Integral
We have \( S = 2\pi \int_{0}^{\sqrt{2}} \sqrt{2(x^2 + 1)} \, dx = 2\pi \int_{0}^{\sqrt{2}} \sqrt{2} \sqrt{x^2 + 1} \, dx = 2\pi \sqrt{2} \int_{0}^{\sqrt{2}} \sqrt{x^2 + 1} \, dx \).
06
Evaluate the Integral
The integral \( \int \sqrt{x^2 + 1} \, dx \) evaluates to a known function. We find \( \int_{0}^{\sqrt{2}} \sqrt{x^2 + 1} \, dx \) using substitution or look up to get \( \int_{0}^{\sqrt{2}} (\frac{1}{2}x\sqrt{x^2+1}+\frac{1}{2}\ln|x+\sqrt{x^2+1}|) \, dx \). Evaluate it at the bounds \( 0 \) and \( \sqrt{2} \).
07
Calculate the Final Surface Area
Plug in \( x = \sqrt{2} \) and \( x = 0 \) to the evaluated integral from Step 6 and substitute back into the formula for \( S \) to find the numerical value of the surface area. Compute the difference.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus Integration
When dealing with surface area of revolution, calculus integration plays a crucial role. It involves the integral of a function over a specified interval. In this scenario, given a curve, we need to revolve it around the x-axis and calculate the area of its surface.
The integration process focuses on summing up infinitesimally small surface area elements along the curve. We employ the formula: \[ S = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \].
This formula integrates the function, multiplied by the derivative's effect, along the x-values from \( a \) to \( b \).
The integration process focuses on summing up infinitesimally small surface area elements along the curve. We employ the formula: \[ S = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \].
This formula integrates the function, multiplied by the derivative's effect, along the x-values from \( a \) to \( b \).
- Essentially, it captures how the curve's slope alters the surface area when it's rotated around the x-axis.
- This formula accurately finds the accumulated area, forming a fuller picture of the object's surface.
Revolving Curves
Revolving curves around an axis transforms the two-dimensional shape into a three-dimensional object. In this case, the curve given by the function \( y = \sqrt{x^2 + 2} \) over the interval \( 0 \leq x \leq \sqrt{2} \) is revolved around the x-axis. This action generates a 3D surface whose area is what we aim to calculate.
Let's delve deeper with some key points:
Let's delve deeper with some key points:
- The chosen axis of revolution determines how the curve turns into a surface.
- As the curve spins, each point traces out a circle, contributing to the overall surface area.
- Using integration, we sum up these differential ring-shaped areas to find the total surface area.
Derivative Calculation
Calculating a derivative accurately is crucial, especially when dealing with surface area of revolution formula. We begin by differentiating the given function \( y = \sqrt{x^2 + 2} \). Using the chain rule, the derivative, \( \frac{dy}{dx} \) is found to be \( \frac{x}{\sqrt{x^2 + 2}} \).
Here's why this matters:
Here's why this matters:
- The derivative \( \frac{dy}{dx} \) feeds into the surface area formula within the square root term, impacting the overall calculation.
- It helps in determining the slope of the curve at any point, providing insights into how it will stretch or compress the surface area as it revolves.
- Accurate derivative calculation ensures the integration performed later is correct and the final surface area isn't underestimated or overestimated.
