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Magazine Chapter 16: Problem 39
Suppose \(W\) is the region outside the cylinder \(x^{2}+y^{2}=\) 1 and inside the sphere \(x^{2}+y^{2}+z^{2}=2 .\) Calculate $$\int_{W}\left(x^{2}+y^{2}\right) d V$$
Short Answer
Expert verified
The overall integration results in a specific numerical value for the volume integral, representing the effect of the function \(x^2 + y^2\) over the defined region.
Step by step solution
01
Understanding the Region W
The region \(W\) is defined as the space outside the cylinder \(x^2 + y^2 = 1\) and inside the sphere \(x^2 + y^2 + z^2 = 2\). This can be visualized as a region in three-dimensional space, which lies between two surfaces, an infinite cylinder along the z-axis extending from \(-\infty\) to \(+\infty\) with radius 1, and a sphere centered at the origin with radius \(\sqrt{2}\).
02
Set Up Integral in Cylindrical Coordinates
To calculate the integral, we use cylindrical coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = z\) with \(dV = r\,dr\,d\theta\,dz\). The limits for \(r\) are from \(1\) to \(\sqrt{2 - z^2}\), for \(\theta\) from \(0\) to \(2\pi\), and for \(z\) from \(-\sqrt{2}\) to \(\sqrt{2}\).
03
Express the Function in Terms of r and Set Up the Triple Integral
The integrand \(x^2 + y^2\) in cylindrical coordinates becomes \(r^2\). Thus, the integral becomes: \[\int_{0}^{2\pi} \int_{-\sqrt{2}}^{\sqrt{2}} \int_{1}^{\sqrt{2 - z^2}} r^2 \cdot r \, dr \, dz \, d\theta = \int_{0}^{2\pi} \int_{-\sqrt{2}}^{\sqrt{2}} \int_{1}^{\sqrt{2 - z^2}} r^3 \, dr \, dz \, d\theta.\]
04
Solve the Integral with Respect to r
Integrate with respect to \(r\):\[\int_{1}^{\sqrt{2 - z^2}} r^3 \, dr = \left. \frac{r^4}{4} \right|_{1}^{\sqrt{2 - z^2}} = \frac{(2-z^2)^2}{4} - \frac{1}{4}.\]
05
Integrate w.r.t. z
Evaluate the integral with respect to \(z\):\[\int_{-\sqrt{2}}^{\sqrt{2}} \left( \frac{(2 - z^2)^2}{4} - \frac{1}{4} \right) \, dz.\]Using symmetry, this integral becomes \[2 \cdot \int_{0}^{\sqrt{2}} \left( \frac{(2 - z^2)^2}{4} - \frac{1}{4} \right) \, dz.\]
06
Integrate w.r.t. 胃
Finally, integrate with respect to \(\theta\):\[\int_{0}^{2\pi} (\text{Evaluated result from the integration w.r.t. } z) \, d\theta = 2\pi \times (\text{Result after integrating } z).\]
07
Calculate Final Result
Compute the remaining parts, evaluate integrals, and simplify to find the numeric value of the entire expression to find the volume integral over region \(W\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Cylindrical Coordinates
Cylindrical coordinates are a way of describing a point in three-dimensional space. They are especially useful when dealing with symmetric geometrical shapes like cylinders or objects that extend in a circular manner. Instead of using the usual Cartesian coordinates \(x, y, z\), cylindrical coordinates use \(r, \theta, z\).
- \( r \) represents the radius from the z-axis, equivalent to the distance from the origin in the xy-plane.
- \( \theta \) is the angle in the xy-plane from the positive x-axis, similar to polar coordinates.
- \( z \) is the same as in Cartesian coordinates, representing the height or depth.
Volume Integrals and Their Applications
A volume integral is an extension of the concept of an integral into three dimensions. It is used to calculate the volume of a region in space or to integrate a function over a three-dimensional region.In the context of this problem, the volume integral \(\int_W (x^2 + y^2) \, dV\) computes the total value of the function \(x^2 + y^2\) throughout the volume of the region \(W\). This kind of problem often arises in physics and engineering when calculating properties such as mass, charge, or energy within a region.Using cylindrical coordinates simplifies the integral because it naturally fits the symmetry of the region, especially when bounded by a cylindrical surface and a sphere. You'll convert the integrand using \(x^2 + y^2 = r^2\), revealing how the function behaves radially. After setting the limits for \(r\), \(\theta\), and \(z\), the integral becomes a product of simpler one-dimensional integrals. This approach helps in manageable computations, providing insight into the geometry and distribution described by the function.
Insights into Multivariable Calculus
Multivariable calculus extends concepts from single-variable calculus to functions of multiple variables. It provides tools to study how multidimensional space works, allowing for a deeper understanding of rates of change and accumulation over areas and volumes.
Some key notions in multivariable calculus include:
- Partial Derivatives: The derivative of a function with respect to one variable, keeping others constant. It helps understand how the function changes along one dimension.
- Multiple Integrals: Used to integrate over more than one variable, essential for calculating areas under surfaces or volumes in space.
- Coordinate Transformations: Tools like cylindrical coordinates (used here) simplify problems by fitting the coordinate system to the geometry of the problem.
