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Magazine Chapter 7: Problem 24
Use a graph to find approximate \(x\) -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the \(x\) -axis the region bounded by these curves. $$y=3 \sin \left(x^{2}\right), \quad y=e^{x / 2}+e^{-2 x}$$
Short Answer
Expert verified
Plot the curves, find intersection points, and compute the integral for volume.
Step by step solution
01
Graph the Curves
First, plot the given functions on the same coordinate system. The first function is \( y = 3 \sin(x^2) \) and the second function is \( y = e^{x/2} + e^{-2x} \). Use graphing software or a graphing calculator to observe where these two curves intersect. Look for points where the curves visually cross over each other.
02
Estimate Intersection Points
From the graph, identify the approximate \(x\)-coordinates where the curves intersect. These will be the bounds for the volume you will calculate later. Typically, you will find moments where the two curves intersect by observing changes or crossing points in the plotted graph.
03
Set Up the Integral for Volume
Using the identified bounds from the intersection points, set up the integral to find the volume of the solid obtained by rotation. Use the formula for volumes of revolution about the x-axis: \[V = \pi \int_{a}^{b} \left( f(x)^2 - g(x)^2 \right) \,dx\]where \( f(x) = 3 \sin(x^2) \) and \( g(x) = e^{x/2} + e^{-2x}. \) Determine \(a\) and \(b\), your limits of integration, from the graph.
04
Compute the Integral
Use a calculator or computer software to evaluate the integral with the limits determined from the intersection points in the previous step. This computation will give you the approximate volume of the solid when this region is revolved around the x-axis.
05
Interpret the Result
The result from the integral calculation is the approximate volume of the generated solid. Ensure the units and context match the problem's requirements.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Intersection of Curves
When working with problems involving the intersection of curves, it's essential first to visualize the problem. Here, you have two functions: a trigonometric function, \( y = 3 \sin(x^2) \), and an exponential function, \( y = e^{x/2} + e^{-2x} \). Finding where these curves intersect involves looking for points where they have the same value. These points of intersection are crucial as they form the bounds for calculating the volume of a solid of revolution.
To identify intersection points:
To identify intersection points:
- First, graph both functions on the same set of axes.
- Visually inspect the graph for where the curves cross over or meet.
- The x-coordinates of these intersection points are used as limits in the integration process.
Graphing Functions
Graphing functions allows for a visual representation of mathematical equations, making it easier to understand their behavior and relationships. In this exercise, graphing the functions \( y = 3 \sin(x^2) \) and \( y = e^{x/2} + e^{-2x} \) is your first step in solving the problem.
Here's how you can do this effectively:
Here's how you can do this effectively:
- Use graphing software or a graphing calculator capable of plotting trigonometric and exponential functions.
- Ensure that the viewing window covers a sufficient range of x-values to see all intersection points.
- The graph should clearly display the shape of each curve as well as their crossing points.
Numerical Integration
Numerical integration involves calculating an integral that cannot easily be solved analytically. In tasks requiring the volume of revolution, after identifying intersection points as limits of integration, you apply numerical methods to evaluate the integral.
The task involves setting up the integral using the formula:\[V = \pi \int_{a}^{b} \left( f(x)^2 - g(x)^2 \right) \,dx\]where \( f(x) = 3 \sin(x^2) \) and \( g(x) = e^{x/2} + e^{-2x} \). Use these steps:
The task involves setting up the integral using the formula:\[V = \pi \int_{a}^{b} \left( f(x)^2 - g(x)^2 \right) \,dx\]where \( f(x) = 3 \sin(x^2) \) and \( g(x) = e^{x/2} + e^{-2x} \). Use these steps:
- Substitute the intersection points as limits \(a\) and \(b\).
- Use technological tools like calculators or software for computation, especially with complex integrands.
Trigonometric Functions
In this exercise, the trigonometric function \( y = 3 \sin(x^2) \) plays a crucial role. Trigonometric functions involve angles and the relationships between the sides of triangles. Here, the argument of the sine function is squared, resulting in a non-standard wave-like curve.
Key points about this function:
Key points about this function:
- The sine function oscillates between -1 and 1, scaled by 3 in this case, meaning it ranges from -3 to 3.
- Squaring \(x\) modifies its typical periodic pattern, resulting in stretching or compressing along the x-axis.
- This function, when graphed, will display peaks and troughs, indicating changes in values.
Exponential Functions
Exponential functions, such as \( y = e^{x/2} + e^{-2x} \), exhibit a distinctive growth or decay pattern. They play a critical role in many real-world applications due to their property of exponential increase or decrease.
Here are some features of these functions:
Here are some features of these functions:
- \( e^{x/2} \) illustrates exponential growth as \(x\) increases, while \( e^{-2x} \) portrays exponential decay, rapidly decreasing as \(x\) increases.
- Combining these functions results in a complex curve that retains characteristics of both growth and decay.
- The resulting curve can have asymptotic behavior, approaching a baseline but never quite intersecting it.
