Question

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-r^{2}}+b} r d z d r d \theta $$

Short answer

The region is a cylinder of radius \( a \) with spherical caps, raised by height \( b \).

Step-by-step solution

Understanding the Order of Integration

The given triple integral is over the variables \( \theta \), \( r \), and \( z \) in cylindrical coordinates. The order of integration indicates that \( \theta \) is in the outermost integral, \( r \) in the middle, and \( z \) in the innermost. This means, for each fixed \( \theta \), we visualize the corresponding sector by varying \( r \) and then \( z \).

Analyzing \( \theta \) Bounds

The limits for \( \theta \) are from 0 to \( 2\pi \). This represents a full rotation around the \( z \)-axis, covering the entire cylindrical region.

Analyzing \( r \) Bounds

The bounds for \( r \) range from 0 to \( a \). This indicates a cylinder with radius \( a \) extending from the center of rotation (the \( z \)-axis) to a radius of \( a \).

Analyzing \( z \) Bounds

The bounds for \( z \) are from 0 to \( \sqrt{a^2 - r^2} + b \). The term \( \sqrt{a^2 - r^2} \) suggests cap-like shapes, as this describes a semisphere located above each point in the base cylinder. Adding \( b \) raises these spherical caps by a height \( b \) along the \( z \)-axis.

Combining the Regions

Combining the bounds, the region consists of a solid body obtained by rotating the base circle of radius \( a \) around the \( z \)-axis where each slice along \( z \) is part of a sphere (due to \( \sqrt{a^2 - r^2} \)) placed on a cylindrical base where \( x^2 + y^2 \leq a^2 \), and then raised by a height \( b \).

Key concepts

Understanding the Triple Integral

A triple integral is a powerful tool used in calculus to evaluate a volume under a three-dimensional region. It extends the idea of a double integral to three dimensions by integrating with respect to three variables, often denoted as \( x \), \( y \), and \( z \), or, in the case of cylindrical coordinates, \( r \), \( \theta \), and \( z \). This approach allows for the calculation and analysis of more complex volumes and regions in space. In the given exercise, the triple integral uses cylindrical coordinates, simplifying the integration over a region that is symmetrical around an axis.In our specific example, the integral is designed to find the volume of a region above a circular base and beneath a cap. Each segment of this volume is accumulated by evaluating the smallest portion of the region, as defined by the integration bounds, and summing up all these portions as we integrate.

Coordinate System Tailored for Cylindrical Regions

A coordinate system is a framework we use to define the position of a point in space. Cylindrical coordinates are one such system, ideal for regions with circular symmetry. Unlike Cartesian coordinates which use \( x \), \( y \), and \( z \) as directions for positioning, cylindrical coordinates deal with three variables: \( r \), \( \theta \), and \( z \).

• \( r \): the radial distance from the \( z \)-axis.
• \( \theta \): the angle formed with the positive \( x \)-axis.
• \( z \): the height along the \( z \)-axis.

By translating the problem to cylindrical coordinates, we make calculations easier for circular or cylindrical shapes, as each point in the space is represented with more intuitive variables for these geometries.

Exploring Geometry in Space

Geometry in space pertains to the study and calculation of properties of spatial figures. In the given problem, the geometry in question forms a solid region defined by specific bounds in the cylindrical coordinate system. The spatial geometry combines elements of both a cylinder and a spherical cap:- The cylinder has a base defined by the circle \( x^2 + y^2 \leq a^2 \), rotating entirely around the \( z \)-axis.- Each segment at a fixed \( r \) is capped by a spherical top defined by \( \sqrt{a^2 - r^2} + b \), creating a solid with a dome-like shape on top.This geometric understanding helps us visualize the region in space more clearly, giving insight into the nature of the solid described by the integral.

Defining the Cylindrical Region

The given triple integral encompasses a specific cylindrical region in space. This region is characterized by:- A full rotation around the \( z \)-axis, as denoted by \( \theta \) ranging from 0 to \( 2\pi \).- A radius~\( r \) that stretches from the center \( 0 \) out to \( a \), forming a circular base at each \( z \) level.- A height defined by \( z \) from \( 0 \) to \( \sqrt{a^2 - r^2} + b \), allowing for a raised cap above a standard cylinder, introducing curvature to the geometry.This special region looks like a solid cylinder topped with a cap that arches upwards, the top being higher than a flat cylinder would reach by the added constant \( b \). Understanding this helps in better visualizing the shape which is crucial for evaluating its volume accurately.

Mastering the Integration Order

The integration order in any multiple integral provides the roadmap for evaluating the integral correctly. It indicates the sequence in which variables are considered and integrated. In our example, we integrate with respect to \( z \), then \( r \), and finally \( \theta \).

• The innermost integration \( \int r \, dz \) concerns the height and the radial part, collecting slices along \( z \).
• Next, \( \int \, dr \) accounts for radial distances, summing all height slices within fixed radial bounds.
• The outermost \( \int \, d\theta \) completes the whole rotation around the \( z \)-axis, essentially rotating the accumulated slices.

This order of integration provides a systematic methodology for approaching complex volume calculations, ensuring each piece of the problem is tackled in a logical progression, critical when working with cylindrical regions.