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Chapter 16: 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 coordinate system.

Step by step solution

01

Understand the expression \(x^{2}+y^{2}+z^{2}\)

In three-dimensional space, the expression \(x^{2}+y^{2}+z^{2}\) can be thought of as the square of the distance, \(r^{2}\), from a point \((x, y, z)\) to the origin \((0, 0, 0)\). Considering the distance from the origin is crucial when choosing a coordinate system that simplifies integration.
02

Recall different coordinate systems in 3D space

There are three common coordinate systems in three-dimensional space: 1. Cartesian coordinates: \((x, y, z)\) 2. Cylindrical coordinates: \((r, \theta, z)\), where \(r\) is the radial distance from the z-axis, and \(\theta\) is the counterclockwise angle measured from the positive x-axis. 3. Spherical coordinates: \((\rho, \phi, \theta)\), where \(\rho\) is the radial distance from the origin, \(\phi\) is the angle between positive z-axis and the line segment connecting the origin to the point, and \(\theta\) is the counterclockwise angle measured from the positive x-axis.
03

Identify the coordinate system best suited for the given integrand

Given the expression \(x^{2}+y^{2}+z^{2}\), we need a coordinate system that simplifies the integrand based on the geometric characteristic of distances from the origin. In this case, the spherical coordinate system is the most suitable choice. In spherical coordinates, \(x^{2}+y^{2}+z^{2}\) is represented by \(\rho^{2}\). This simplifies the integrand and often makes the triple integral easier to compute.

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Most popular questions from this chapter

Evaluate the following integrals using polar coordinates. Assume \((r, \theta)\) are polar coordinates. A sketch is helpful. $$\iint_{R}\left(x^{2}+y^{2}\right) d A ; R=\\{(r, \theta): 0 \leq r \leq 4,0 \leq \theta \leq 2 \pi\\}$$

Consider the following two- and three-dimensional regions with variable dimensions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A solid cone has a base with a radius of \(a\) and a height of \(h .\) How far from the base is the center of mass?

Find the volume of the solid bounded by the surface \(z=f(x, y)\) and the \(x y\)-plane. (Check your book to see figure) $$f(x, y)=e^{-\left(x^{2}+y^{2}\right) / 8}-e^{-2}$$

Improper integrals 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: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} g(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} g(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{5 / 2}} ; R=\\{(r, \theta): 1 \leq r<\infty, 0 \leq \theta \leq 2 \pi\\}$$

Solids bounded by paraboloids Find the volume of the solid below the paraboloid \(z=4-x^{2}-y^{2}\) and above the following polar rectangles. $$R=\\{(r, \theta): 0 \leq r \leq 2,0 \leq \theta \leq 2 \pi\\}$$
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