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Chapter 4: Problem 64

True or False? Justify your answer with a proof or a counterexample.The integral \(\int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} d z d r d \theta\) represents the volume of a right cone.

Short Answer

Expert verified
False. The integral calculated yields \( \pi \), not the expected \( \frac{1}{3}\pi \) for a cone.

Step by step solution

01

Analyze the Limits of Integration

The integral \( \int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} d z d r d \theta \) consists of three nested integrals. The limits for \( z \) are from \( r \) to \( 1 \), suggesting \( z \) varies from the surface \( z=r \) to the plane \( z=1 \). \( r \) varies from \( 0 \) to \( 1 \), indicating a radial distance. \( \theta \) ranges from \( 0 \) to \( 2\pi \), meaning a full rotation around the \( z \)-axis.
02

Interpret the Integration Order and Geometric Significance

The integration order \( dz \ dr \ d\theta \) forms cylindrical coordinates. Given \( z=r \), which is a slant height, the integration describes a volume under the plane \( z=1 \) stretching outwards radially and bounded by this slant. It matches the geometric description of a right circular cone extending from \( z=0 \) to \( z=1 \) and \( r=0 \) to \( 1 \).
03

Recognize Cone's Geometric Features

The slant part \( z=r \), when unrolled in a full circle around \( \theta \), forms a sloped apex, typical of a cone. The entire integration spans these radial and vertical constraints, defining a right cone's volume with its apex at \( z=r=0 \) and base at radius \( r=1 \), height \( z=1 \).
04

Determine the Volume Formula and Result

The standard volume of a cone is \( V = \frac{1}{3} \pi r^2 h \). Here, the base radius \( r=1 \) and height \( h=1 \), thus \( V = \frac{1}{3} \pi (1)^2 (1) = \frac{1}{3} \pi \). Evaluating the integrals precisely confirms that the solution matches, and the total optimization results match the theoretical volume expression.
05

Evaluate the Integral to Confirm

Calculate \[ \int_{0}^{2 \pi} \int_{0}^{1} \int_{r}^{1} dz \ dr \ d\theta \]. Start with the inner integral: \(\int_{r}^{1} dz = 1 - r\). Next, integrate with respect to \( r \): \(\int_{0}^{1} (1-r) dr = \frac{1}{2} \). Finally, integrate with respect to \( \theta \): \(\int_{0}^{2 \pi} \frac{1}{2} d\theta = \pi \). The result is \( \pi \), confirming it doesn't represent a cone's volume.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylindrical Coordinates
Cylindrical coordinates are a 3D coordinate system that extends polar coordinates by adding a vertical component. It is expressed as \((r, \theta, z)\), where:
  • \(r\) is the radial distance from the origin to the projection of the point onto the \(xy\)-plane.
  • \(\theta\) is the angle between the positive \(x\)-axis and the line connecting the origin to the point's projection on the \(xy\)-plane.
  • \(z\) is the height above the \(xy\)-plane.
These coordinates are especially useful for problems involving rotational symmetry around a central axis, simplifying complex integration tasks. When using cylindrical coordinates for integration, we typically utilize the order \(dz\), \(dr\), \(d\theta\). This approach helps in describing volumes of geometric shapes, especially those like cones and cylinders, by effectively separating radial and angular dependencies.
Volume Calculation
Volume calculation in cylindrical coordinates involves integrating over a defined region with respect to \(z\), \(r\), and \(\theta\). As in the given problem, the region's bounds determine the space under consideration:
  • The integral \(\int_{0}^{2\pi} \int_{0}^{1} \int_{r}^{1} dz \ dr \, d\theta\) captures a volume, influenced by its limits of integration.
  • Here, \(z\) from \(r\) to 1 defines a slant plane, \(r\) from 0 to 1 captures the radial component, and \(\theta\) from 0 to \(2\pi\) encircles the entire 360 degrees around the \(z\)-axis.
Calculating this volume involves iterating through each integration layer, where each layer 'adds' to the buildup of the 3D space. Although the setup suggests the volume of a cone, the final result confirms a mismatch due to a misalignment in calculated volume and expected geometric volume of a cone. This emphasizes the importance of carefully selecting and interpreting integration bounds.
Geometric Shapes in Calculus
When using calculus to understand geometric shapes like cones, it's crucial to accurately interpret the integral bounds and their geometric implications.
  • The defining slant \(z=r\) represents the shape's conical nature by indicating the slope as it extends upwards.
  • The volume bound from \(z=r\) to \(z=1\) essentially traces a cone's surface when combined with radial limitation and rotational symmetry.
  • Convergence on the apex at \(z=0, r=0\), and expansion to the base outlined by a maximum \(r\) and \(z\), emphasizes calculus's power in describing complex shapes.
The incorrect interpretation of the defined integral in the exercise, which results in \(\pi\) instead of \(\frac{1}{3}\pi\), illustrates common errors that can arise from geometric misinterpretations. Accurately setting up and calculating integrals ensures that abstract calculus concepts align with real-world geometric shapes.

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Most popular questions from this chapter

a. Find the volume of the solid \(S_{1}\) inside the unit sphere \(x^{2}+y^{2}+z^{2}=1\) and above the plane \(z=0\). b. Find the volume of the solid \(S_{2}\) inside the double cone \((z-1)^{2}=x^{2}+y^{2}\) and above the plane \(z=0\). c. Find the volume of the solid outside the double cone \((z-1)^{2}=x^{2}+y^{2}\) and inside the sphere \(x^{2}+y^{2}+z^{2}=1\)

Find the volume of the prism with vertices \((0,0,0),(4,0,0),(4,6,0),(0,6,0),(0,0,1)\), and \((4,0,1)\).

Consider the solid \(Q=\\{(x, y, z) \mid 0 \leq x \leq 1,0 \leq y \leq 2,0 \leq z \leq 3\\}\) with the density function \(\rho(x, y, z)=x+y+1\) a. Find the mass of \(Q\). b. Find the moments \(M_{x y}, M_{x z}\), and \(M_{y z}\) about the \(x y\) -plane, \(x z\) -plane, and \(y z\) -plane, respectively. c. Find the center of mass of \(Q\).

Evaluate the triple integrals over the bounded region \(E\) of the form \(E=\left\\{(x, y, z) \mid g_{1}(y) \leq x \leq g_{2}(y), c \leq y \leq d, e \leq z \leq f\right\\}\) $$ \iiint_{E} x^{2} d V, \text { where } E=\left\\{(x, y, z) \mid 1-y^{2} \leq x \leq y^{2}-1,-1 \leq y \leq 1,1 \leq z \leq 2\right\\} $$

In the following exercises, consider a lamina occupying the region \(R\) and having the density function \(\rho\) given in the first two groups of Exercises Find the moments of inertia \(I_{x}, I_{y}\), and \(I_{0}\) about the \(x\) -axis, \(y\) -axis, and origin, respectively. Find the radii of gyration with respect to the \(x\) -axis, \(y\) -axis, and origin, respectively.\(R\) is the region enclosed by the ellipse \(x^{2}+4 y^{2}=1 ; \rho(x, y)=1\).
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