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Magazine Chapter 7: Problem 19
Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. $$y=e^{-x^{2}}, y=0, x=-1, x=1$$ (a) About the \(x\) -axis (b) About \(y=-1\)
Short Answer
Expert verified
(a) Volume about the x-axis is 2.77296 units鲁; (b) Volume about y=-1 is 10.20867 units鲁.
Step by step solution
01
Understand the Problem
We need to find the volume of the solid formed by rotating the region bounded by the curves \( y = e^{-x^2}, y = 0, x = -1, \) and \( x = 1 \). Specifically, we're asked to perform this rotation about two different lines: the \( x \)-axis and \( y = -1 \).
02
Set up the integral for rotation about the x-axis
For rotation about the \( x \)-axis, we use the formula for the volume of solids of revolution:\[ V = \pi \int_a^b [f(x)]^2 \, dx \]Where \( f(x) = e^{-x^2} \) and the limits of integration are \( x = -1 \) to \( x = 1 \). Thus, the integral becomes:\[ V = \pi \int_{-1}^{1} (e^{-x^2})^2 \, dx = \pi \int_{-1}^{1} e^{-2x^2} \, dx \]
03
Calculate the integral for rotation about the x-axis
Evaluating the integral \( \pi \int_{-1}^{1} e^{-2x^2} \, dx \) requires a calculator with numerical integration capabilities. Enter this expression into the calculator to find its value to five decimal places, yielding:\[ V = \pi \times 0.88208 \approx 2.77296 \] cubic units.
04
Understand the second rotation scenario
Now we set up an integral for the volume of the solid formed by rotating the same region about the line \( y = -1 \). This requires using the shell method, as it rotates around a horizontal line below the \( x \)-axis.
05
Set up the integral for rotation about y = -1
Using the shell method, the volume \( V \) is given by:\[ V = 2\pi \int_a^b (r(x))(h(x)) \, dx \]Where \( r(x) = e^{-x^2} + 1 \) (distance from the axis \( y = -1 \)), and \( h(x) = e^{-x^2} \). The limits remain \( x = -1 \) to \( x = 1 \):\[ V = 2\pi \int_{-1}^{1} (e^{-x^2} + 1)e^{-x^2} \, dx = 2\pi \int_{-1}^{1} (e^{-2x^2} + e^{-x^2}) \, dx \]
06
Calculate the integral for rotation about y = -1
Evaluate the integral \( 2\pi \int_{-1}^{1} (e^{-2x^2} + e^{-x^2}) \, dx \) using a calculator. This yields:\[ V = 2\pi(0.88208 + 0.74682) \approx 10.20867 \] cubic units.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Volumes of Revolution
Calculating the volumes of revolution involves finding the volume of a solid formed by revolving a region around a specified axis or line. To understand this better:
- We define a region bounded by certain curves; in this case, it's the area under the curve of the function between given boundaries.
- This region is then rotated around a line (e.g., the x-axis or another horizontal/vertical line) to create a 3D shape.
- The main goal is to calculate the volume of this resulting solid.
Solids of Revolution
Solids of revolution are the 3D shapes created by rotating 2D regions around a line (the axis of rotation). You deal with these when solving calculus problems that involve integration, like the one in the original exercise. Here's a closer look:
- The simplest form is when you have a curve, like a parabola or sine wave, rotating around an axis close to the curve.
- The resulting solid鈥檚 volume can be calculated by understanding the cross-sections perpendicular to the axis of rotation.
- In many cases, these cross-sections take the shape of disks or washers, leading to the use of the disk or washer method of integration.
Shell Method
The shell method is an integration technique used for finding the volume of solids of revolution, especially when rotating around an axis not a boundary of the function, like when the axis is horizontal and below the function. Here鈥檚 how it typically works:
- Identify the radius and height for each cylindrical shell relative to the axis of rotation.
- Each cylindrical shell contributes a small volume, calculated as the product of circumference, height, and thickness. The formula is given by: \( V = 2\pi \int_a^b (r(x))(h(x)) \, dx \).
- This method is particularly useful when the axis of rotation is parallel to the x-axis and somewhere other than y=0 or the x-axis itself.
Numerical Integration
When dealing with complex functions that cannot be integrated easily using elementary methods, numerical integration becomes crucial. It involves approximate calculations, often aided by computers or calculators capable of performing such operations. Here's what happens:
- Numerical integration techniques include methods like the Trapezoidal Rule, Simpson鈥檚 Rule, and others, which are algorithms for approximating the values of definite integrals.
- These techniques divide the integral into subintervals and calculate areas under the curve approximately.
- In the example provided, calculating integrals of exponential functions like \( e^{-2x^2} \) requires numerical methods, as they can't be expressed with simple antiderivatives.
