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Chapter 13: Problem 10

What coordinate system is suggested if the integrand of a triple integral involves \(x^{2}+y^{2}+z^{2} ?\)

Short Answer

Expert verified
Answer: Spherical Coordinates (\(\rho, \theta, \phi\)).

Step by step solution

01

Evaluate the integrand

The integrand involves the sum of squares of the variables: \(x^2+y^2+z^2\). It's worth noting that this expression embodies the square of the euclidean distance from the origin to the point (x, y, z) in a Cartesian coordinate system.
02

Compare with coordinate systems

Now, let's recall the main features and formulas for each coordinate system: 1. Cartesian Coordinates: \((x, y, z)\) 2. Cylindrical Coordinates: \((r, \theta, z)\) where \(r = \sqrt{x^2 + y^2}\), \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = z\). 3. Spherical Coordinates: \((\rho, \theta, \phi)\) where \(\rho = \sqrt{x^2 + y^2 + z^2}\), \(x = \rho\sin\phi\cos\theta\), \(y = \rho\sin\phi\sin\theta\), and \(z = \rho\cos\phi\).
03

Identify the suitable coordinate system

Notice that the integrand \(x^2 + y^2 + z^2\) is equivalent to \(\rho^2\) in spherical coordinates. This means that spherical coordinates are likely a more natural choice, as the integrand simplifies to a single term. In cylindrical coordinates, the integrand becomes \(r^2 + z^2\), which is still a sum of squares. This is less suitable than spherical coordinates, but potentially more suitable than Cartesian coordinates.
04

Conclusion

Based on the integrand \(x^2 + y^2 + z^2\), the suggested coordinate system would be Spherical Coordinates, represented by \((\rho, \theta, \phi)\).

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Most popular questions from this chapter

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Find the center of mass of the upper half of \(D(z \geq 0)\) assuming it has a constant density.

Determine whether the following statements are true and give an explanation or counterexample. a. Any point on the \(z\) -axis has more than one representation in both cylindrical and spherical coordinates. b. The sets \(\\{(r, \theta, z): r=z\\}\) and \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) are the same.

Improper integrals arise in polar coordinates when the radial coordinate \(r\) becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} f(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} f(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\iint_{R} e^{-x^{2}-y^{2}} d A ; R=\\{(r, \theta): 0 \leq r < \infty, 0 \leq \theta \leq \pi / 2\\}$$

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. That part of the ball \(\rho \leq 2\) that lies between the cones \(\varphi=\pi / 3\) and \(\varphi=2 \pi / 3\)

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The solid inside the sphere \(\rho=1\) and below the cone \(\varphi=\pi / 4\) for \(z \geq 0\)
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