Question

Use cylindrical coordinates to find the indicated quantity. Volume of the solid bounded above by the plane \(z=y+4\), below by the \(x y\) -plane, and laterally by the right circular cylinder having radius 4 and whose axis is the \(z\) -axis.

Short answer

The volume of the solid is \(64\pi\).

Step-by-step solution

Convert to Cylindrical Coordinates

Cylindrical coordinates are given by \( (r, \theta, z) \), where \(x = r \cos\theta\), \(y = r \sin\theta\), and \(z = z\). The equation on top is \(z = y + 4\), which converts to \(z = r \sin\theta + 4\) in cylindrical coordinates. The lateral boundary is given by the cylinder \(r = 4\).

Set up the Volume Integral

The solid is bounded below by the \(xy\)-plane \((z=0)\), laterally by the cylinder \(r=4\), and above by the plane \(z = r \sin\theta + 4\). The volume can be found by integrating: \[ V = \int_{0}^{2\pi} \int_{0}^{4} \int_{0}^{r\sin\theta+4} r \, dz \, dr \, d\theta. \]

Integrate with Respect to \(z\)

Integrate the innermost integral with respect to \(z\): \[ \int_{0}^{r\sin\theta+4} r \, dz = r[z]_{0}^{r\sin\theta+4} = r(r\sin\theta+4). \]

Integrate with Respect to \(r\)

Substitute \(r(r\sin\theta+4)\) into the integral and integrate with respect to \(r\): \[ \int_{0}^{4} r(r\sin\theta+4)\, dr = \int_{0}^{4} (r^2 \sin\theta + 4r)\, dr. \]Perform the integration:\[ \left[ \frac{r^3}{3} \sin\theta + 2r^2 \right]_{0}^{4} = \left[ \frac{64}{3}\sin\theta + 32 \right]. \]

Integrate with Respect to \(\theta\)

Integrate with respect to \(\theta\) over the interval \([0, 2\pi]\):\[ \int_{0}^{2\pi} \left( \frac{64}{3} \sin\theta + 32 \right) \, d\theta. \]The integral of \( \frac{64}{3} \sin\theta \) over \([0, 2\pi]\) is zero. So we have:\[ \int_{0}^{2\pi} 32 \, d\theta = 32[\theta]_{0}^{2\pi} = 64\pi. \]

Calculate the Final Volume

Combine the results of the integration to find the total volume. The calculation yields the volume of the solid as \(64\pi\).

Key concepts

Volume Integral

When finding the volume of a three-dimensional solid in a mathematical space, a volume integral is often used. The idea behind a volume integral is to sum up the infinitesimally small volume elements within the boundaries of the solid. This summation gives the total volume of the object.

In our specific exercise, the volume integral is crucial to determine the space occupied by a solid that is bound by a plane on top, a right circular cylinder laterally, and the xy-plane below. We represent this volume with an integral:

• The outermost integral encompasses the entire angular section, ranging from 0 to \(2\pi\).
• The middle integral sweeps through the radial section from the center to the boundary, which is the radius of the cylinder (4 in this problem).
• The innermost integral covers the height, from the base at \(z=0\) to the top plane defined in the problem.

Ultimately, the integration over these dimensions, represented in cylindrical coordinates, gives us the required volume of the solid.

Triple Integration

Triple integration is a technique used when calculating the integral of a function over a three-dimensional region. Through this process, we can find quantities such as volume, mass, or charge within a certain boundary. Each integration corresponds to one of the three spatial dimensions.

In the example from the original exercise, we use triple integration to determine the volume. Here is how the process works:

• First, we integrate with respect to \(z\), which accounts for the height of the solid in every infinitesimally thin disk.
• Next, we integrate with respect to \(r\), tracing the radial distance from the center to the extreme edge of the cylinder.
• Finally, we integrate with respect to \(\theta\), moving around the full circular sweep of the cylinder, thereby enclosing the entire volume.

The result from this triple integration gives us the total space occupied by the solid inside the specified bounds. It is an important tool for solving spatial problems, especially those with complex shapes and constraints.

Coordinate Conversion

Coordinate conversion is an essential concept when dealing with integrals in different coordinate systems. It allows us to transform variables and equations into a form that is easier to work with, especially in irregular or circular geometries like cylinders.

In this particular exercise, converting from Cartesian coordinates

• (x, y, z)
• to cylindrical coordinates(r, \( \theta \), z)
gives us extra flexibility in calculation.

• In cylindrical coordinates:
• \(x = r \cos\theta\)
• \(y = r \sin\theta\)
• \(z = z\)

These new variables allow us to naturally describe objects like right circular cylinders. This conversion simplifies the boundary equations and makes the integration more manageable, leading to the correct evaluation of the volume integral. Using cylindrical coordinates is often the best choice when dealing with problems involving symmetry around an axis.

Right Circular Cylinder

A right circular cylinder is a three-dimensional geometric shape characterized by two parallel bases connected by a curved surface. The bases are circles, and the central axis runs perpendicular through their centers, extending through the entire height of the cylinder.

In the exercise we are working on, the right circular cylinder is given a fixed radius of 4 and centers along the z-axis. Understanding this geometry is crucial in setting up the limits and equations for integration.

• The radius of the cylinder defines the lateral boundary, which simplifies the integration limits in the radial direction.
• The axis of the cylinder, the z-axis, means that using cylindrical coordinates aligns perfectly with the problem's spatial setup.

Knowing how to handle a right circular cylinder is necessary when applying techniques like coordinate conversion and volume integration. These concepts work together to ensure the calculations are both straightforward and accurate, reflecting the exact physical characteristics of the object being analyzed.