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Chapter 16: Problem 9

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

Short Answer

Expert verified
Answer: Cylindrical coordinates.

Step by step solution

01

Recall Coordinate Systems

There are three main coordinate systems: Cartesian (rectangular) coordinates, cylindrical coordinates, and spherical coordinates. Each coordinate system has its own unique way of representing points in space, and different coordinate systems are preferred for different types of problems due to their ability to simplify calculations.
02

Analyze the integrand

We are given a triple integral with an integrand involving the expression \(x^2 + y^2\). The goal is to find a coordinate system that simplifies this expression for easier integration.
03

Evaluate coordinate systems

Let's take a look at each of the three coordinate systems and their most important features: 1. Cartesian (rectangular) coordinates: \((x, y, z)\) - best suited for problems involving rectangular shapes. 2. Cylindrical coordinates: \((\rho, \phi, z)\) - best suited for problems involving cylindrical or circular shapes. 3. Spherical coordinates: \((r, \theta, \phi)\) - best suited for problems involving spherical shapes.
04

Choose the coordinate system

In cylindrical coordinates, we have the following transformation relations: $$ x = \rho \cos \phi \\ y = \rho \sin \phi \\ z = z $$ Notice that, in cylindrical coordinates, \(x^2 + y^2 = \rho^2\cos^2\phi + \rho^2\sin^2\phi = \rho^2(\cos^2\phi + \sin^2\phi) = \rho^2\). In other words, cylindrical coordinates simplify the expression \(x^2 + y^2\) to just \(\rho^2\). Therefore, for a triple integral involving \(x^2 + y^2\), cylindrical coordinates would be the most appropriate choice.

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Most popular questions from this chapter

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. $$\iint_{R} e^{-x^{2}-y^{2}} d A ; R=\left\\{(r, \theta): 0 \leq r<\infty, 0 \leq \theta \leq \frac{\pi}{2}\right\\}$$

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. $$\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{5 / 2}} ; R=\\{(r, \theta): 1 \leq r<\infty, 0 \leq \theta \leq 2 \pi\\}$$

Determine whether the following statements are true and give an explanation or counterexample. a. A thin plate of constant density that is symmetric about the \(x\) -axis has a center of mass with an \(x\) -coordinate of zero. b. A thin plate of constant density that is symmetric about both the \(x\) -axis and the \(y\) -axis has its center of mass at the origin. c. The center of mass of a thin plate must lie on the plate. d. The center of mass of a connected solid region (all in one piece) must lie within the region.

Mass from density data The following table gives the density (in units of \(\mathrm{g} / \mathrm{cm}^{2}\) ) at selected points (in polar coordinates) of a thin semicircular plate of radius \(3 .\) Estimate the mass of the plate and explain your method. $$\begin{array}{|c|c|c|c|c|c|}\hline & \boldsymbol{\theta}=\mathbf{0} & \boldsymbol{\theta}=\pi / 4 & \boldsymbol{\theta}=\pi / 2 &\boldsymbol{\theta}=3 \pi / 4 & \boldsymbol{\theta}=\pi \\\\\hline r=1 & 2.0 & 2.1 & 2.2 & 2.3 & 2.4 \\\\\hline r=2 & 2.5 & 2.7 & 2.9 & 3.1 & 3.3 \\\\\hline r=3 & 3.2 & 3.4 & 3.5 & 3.6 & 3.7 \\\\\hline\end{array}$$

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