Question

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

Short answer

The region is a quarter-cylinder with height 2, radius 2, spanning a quarter-circle in the first quadrant.

Step-by-step solution

Understanding the Integral in Cylindrical Coordinates

The given integral is in cylindrical coordinates and is written as \( \int_{0}^{\pi / 2} \int_{0}^{2} \int_{0}^{2} r \, dz \, dr \, d\theta \). This notation indicates that we are integrating over the region defined by these bounds in the space of cylindrical coordinates \((r, \theta, z)\). It suggests that the region can be described in terms of these three variables and their corresponding limits.

Interpreting Each Variable and Its Bounds

The variable \(r\) represents the radial distance from the \(z\)-axis. Its bounds are \(0\) to \(2\), meaning the region is radially extending from the center up to a distance of \(2\). The variable \(\theta\) represents the angle from the positive \(x\)-axis in the plane, ranging from \(0\) to \(\pi/2\), which specifies a quarter-circle in the \(xy\)-plane. Lastly, \(z\) is the height relative to the \(xy\)-plane, ranging from \(0\) to \(2\), forming the vertical extent of the region.

Determining the Shape of the Region in Space

Considering the bounds in cylindrical coordinates, the region represents a quarter-cylinder. The cylinder extends vertically from \(z=0\) to \(z=2\), radially from the center (\(r=0\)) up to \(r=2\), and covers an angular range from \(\theta=0\) to \(\pi/2\), which is a quarter circle. This results in the region being a quarter of a cylinder within the first octant, considering the constraints of \(r\), \(\theta\), and \(z\).

Describing the Specific Region

The volume described by the integral is therefore a quarter-cylinder, with height \(2\), radius \(2\), and subtending an angle of \(\pi/2\). The quarter-cylinder sits on the \(xy\)-plane and extends vertically upward to \(z=2\), placed in the first octant due to \(\theta\)'s range. It looks like a quarter-doughnut cut out of a cylindrical shape, extending along the positive \(x\) and \(y\) axes.

Key concepts

Triple Integral

In mathematics, a triple integral allows us to calculate the volume under a surface within a three-dimensional region. This integral involves three integrals nested together, each accounting for a different dimension of space. It is similar to calculating the area under a curve with a single integral or the volume under a surface using a double integral, but now expanded to account for all three dimensions.
Triple integrals are essential tools in calculating volumes and other properties of three-dimensional objects. By evaluating these integrals over a defined region, we can determine quantities such as mass, volume, and center of mass, especially when density and other factors vary within the object. Understanding the limits and bounds is crucial as they define the specific region in which the integration is performed.
When using triple integrals, we often switch to more convenient coordinate systems like spherical or cylindrical coordinates, depending on the symmetry and shape of the region of integration. This facilitates easier computation and better alignment with the geometric properties of the object at hand.

Quarter-Cylinder

A quarter-cylinder is a three-dimensional geometric shape representing one-fourth of a full cylindrical shape. Imagine dividing a cylinder into four equal parts by slicing it through the vertical axis and then cutting it again perpendicularly — each portion resembles a quarter-cylinder.
In the context of integration, especially using cylindrical coordinates, considering a quarter-cylinder is convenient. This is due to its symmetry and straightforward boundaries, making it easier to define in mathematical terms.
In this exercise, the quarter-cylinder extends from the central axis (at distance zero) to a radius of 2 units. Its height ranges up to 2 units vertically. Moreover, it spans an angular segment from 0 to \(\pi/2\), which is a quarter of a full circle's 360 degrees. Together, these characteristics describe a three-dimensional volume that occupies the first octant of a Cartesian system.

Cylindrical Integration

Cylindrical integration is a method employed to solve integrals involving regions in space that exhibit cylindrical symmetry. It is a powerful tool for efficiently calculating integrals over regions such as cylinders, cones, and similar shapes.
The key aspect of cylindrical integration is the representation of points in space using cylindrical coordinates \((r, \theta, z)\). Here, \(r\) is the radial distance from the central axis, \(\theta\) is the angular displacement from the reference direction (usually the positive \(x\)-axis), and \(z\) is the height above the \(xy\)-plane.
Cylindrical integration is particularly useful when the region of integration naturally aligns with cylindrical coordinates, thus minimizing the complexity of the limits and boundaries. By integrating with respect to \(z\), \(r\), and \(\theta\) in respective order, we account for the three-dimensional extent of the objects under consideration. This method simplifies integration processes and provides a direct route to determine volumes and related quantities in structures with radial symmetry.

Mathematical Bounds

Mathematical bounds are constraints that indicate the limits of integration for each variable in a given integral. They specify the starting and ending points of integration, crucial for accurately describing the geometric region under consideration.
In the given exercise, the bounds are defined by the variables \(r\), \(\theta\), and \(z\). Here's how each of them contributes:

• The radial distance \(r\) ranges from 0 to 2, setting the radius for the quarter-cylinder.
• The angle \(\theta\) is bounded between 0 and \(\pi/2\), describing a quarter of the rotational symmetry.
• The height \(z\) goes from 0 to 2, establishing the vertical extent of the region.

These bounds transform the integration problem into a precise geometric task of finding volumes and other related entities confined within these specified limits. Understanding and properly setting mathematical bounds is integral to successfully applying triple integrals in real-world situations, ensuring computations align with the intended physical constraints and symmetries.