Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 \((0,3)\)

Step-by-Step Solution

Answer

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 1: 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 \((0,0), (2,0), (2,3), (0,3)\).

Step 2: 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.

Step 3: 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 \([-\sqrt{9-x^2}, \sqrt{9-x^2}]\).

Step 4: 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.

Step 5: 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.

Key Concepts

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—in 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—here, 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—that 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—whether 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.