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Chapter 13: Problem 5

The part of the cylinder \(x^{2}+z^{2}=9\) that is directly over the rectangle in the \(x y\)-plane with vertices \((0,0),(2,0),(2,3)\), and \((0,3)\)

Short Answer

Expert verified
The solution is a cylindrical surface segment over the rectangle with x-values from 0 to 2, y-values from 0 to 3, and z-values based on the equation \(z^2 = 9 - x^2\).

Step by step solution

01

Understanding the Problem

We are given a cylinder with the equation \(x^{2}+z^{2}=9\). This equation describes a cylinder along the y-axis with a radius of 3, since replacing \(x\) and \(z\) gives a circle with radius 3 when sliced by any plane parallel to the xy-plane. We need to find the part of this cylinder that lies directly above a rectangle in the xy-plane with vertices \((0,0), (2,0), (2,3), (0,3)\).
02

Defining the Bounds for x and y

The rectangle in the xy-plane defines the domain for \(x\) and \(y\). The bounds for \(x\) are from 0 to 2, and for \(y\), from 0 to 3.
03

Analyzing Cylinder Constraints

The cylinder's surface is defined by the equation \(x^{2} + z^{2} = 9\). Since \(x\) can range from 0 to 2, substituting it into the cylinder's equation gives possible values for \(z\). For these x-values, \(z\) will vary within \([-\sqrt{9-x^2}, \sqrt{9-x^2}]\).
04

Constructing the Solid

Since \(y\) reflects the depth/direction along the axis of the cylinder, \(z\) is determined only by \(x\) due to the cylinder's equation. The valid z-values for each \(x\) are described by the cylinder's equation. Therefore, for each fixed \(x\) in the interval \([0, 2]\), the \(z\)-coordinate can be anywhere between \(-\sqrt{9-x^2}\) and \(\sqrt{9-x^2}\), and \(y\) can vary from 0 to 3.
05

Calculating the Limits of z When x=0 and x=2

For \(x=0\), \(z^2=9\), thus \(z = \pm 3\). For \(x=2\), \(z^2=5\), thus \(z = \pm \sqrt{5}\). The function \(z=\sqrt{9-x^2}\) decreases from 3 to \(\sqrt{5}\) as \(x\) goes from 0 to 2, defining the top half of the cylinder over the rectangle.

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Key Concepts

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

Cylinder Equation
The cylinder equation given is \(x^2 + z^2 = 9\). This equation represents a 3D surface in space. A cylinder can be thought of as a series of circles stacked along a line鈥攊n this case, the y-axis. When we see an equation in the form \(x^2 + z^2 = R^2\), it's describing a cylinder with a radius \(R\) along the third variable's axis鈥攈ere, the y-axis.
The equation can be visualized as:
  • Each slice parallel to the top circle (the xy-plane) is a circle with radius 3 at any point along the y-axis.
  • The coordinates \((x, z)\) fit into a circle centered at the origin with given radius.
To better understand, consider holding a hula hoop (a circle) straight up and moving it parallel to itself along a straight path鈥攖hat represents how the cylinder equation behaves in space.
Limits of Integration
When tackling any 3D problem involving integration, defining the limits of integration is crucial. In this scenario, we're examining a part of the cylindrical surface directly above a rectangle in the xy-plane. The limits here define the 'area of interest.'
  • The x-value ranges from 0 to 2, based on the rectangle's vertices projections onto the x-axis.
  • For y-values, the range is from 0 to 3.
  • The z-values are determined by the cylinder's restriction: they fluctuate from \(-\sqrt{9-x^2}\) to \(\sqrt{9-x^2}\).
These limits of integration specify the 3D boundaries within which the calculations for the volume or area might occur. They're crucial for quantifying the specified volume鈥攚hether it's for integration or simply conceptually understanding the 'slice' of reality we're working on.
Cylinder Radius
The radius of the cylinder is a fundamental property. This cylinder, defined by \(x^2 + z^2 = 9\), indicates a radius \(R = 3\). Understanding how the radius influences the cylinder is key.
  • The radius determines the size of the circular cross-sections in the xz-plane.
  • Any circle derived within the cylinder, parallel to xz, will always have this radius.
This consistent radius means no matter where we slice the cylinder parallel to the xz-plane, each section reveals a circle with a fixed size. It simplifies finding areas within the cylinder's boundary over any specified range.
XYZ-Coordinate System
The xyz-coordinate system is foundational in visualizing problems involving three dimensions, like this cylinder over a rectangle.
  • The x-axis runs horizontally, typically left to right.
  • The y-axis runs vertically across the plane, often bottom to top.
  • The z-axis comes towards or away from the viewer, providing depth.
In this problem, the cylinder extends along the y-axis, and crucially, it can be thought of as tracing out an area over the xz-plane based on the given equation. The coordinates (x, y, z) help express points on this cylinder, where x and z provide the circle's position at any line y. This three-dimensional view gives clear pictorial representation and maintains spatial relationships throughout mathematical solutions.

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Most popular questions from this chapter

In Problems \(7-14\), use cylindrical coordinates to find the indicated quantity. Volume of the solid bounded above by the sphere \(r^{2}+z^{2}=5\) and below by the paraboloid \(r^{2}=4 z\)

In Problems 13-18, an iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the region. $$ \int_{0}^{\pi} \int_{0}^{\sin \theta} r d r d \theta $$

Suppose that the random variables \(X\) and \(Y\) have joint PDF $$ f(x, y)= \begin{cases}\frac{1}{4}, & \text { if } 0 \leq x \leq 2,0 \leq y \leq 2 \\ 0, & \text { otherwise }\end{cases} $$ that is, \(X\) and \(Y\) are uniformly distributed over the square \(0 \leq x \leq 2,0 \leq y \leq 2\). Find (a) the joint PDF of \(U=X+Y\) and \(V=X-Y\), and (b) the marginal PDF of \(U\).

Find the volume of the solid inside both of the spheres \(\rho=2 \sqrt{2} \cos \phi\) and \(\rho=2\).

In Problems 1-6, evaluate the iterated integrals. $$ \int_{0}^{\pi / 2} \int_{0}^{\sin \theta} r d r d \theta $$
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