Problem 32 - Calculus Early Transcendentals

Question

Use cylindrical coordinates to find the volume of the following solids. The solid cylinder whose height is 4 and whose base is the disk \(\\{(r, \theta): 0 \leq r \leq 2 \cos \theta\\}\)

Step-by-Step Solution

Answer

Answer: The volume of the solid cylinder is \(V = 8 \pi\).

Step 1: Understand cylindrical coordinates

Cylindrical coordinates are a 3D coordinate system represented by \((r, \theta, z)\). Here, \(r\) is the radial distance from the z-axis, \(\theta\) is the angle made by the radial line with the positive x-axis, and \(z\) is the height along the z-axis. The volume element in cylindrical coordinates is given by \(dV = r \, dr \, d\theta \, dz\).

Step 2: Set up the volume integral

To find the volume of the solid cylinder, we integrate the volume element \(dV\) over the cylinder. We need to determine the bounds of integration for \(r\), \(\theta\), and \(z\). Given the base disk equation \(0 \leq r \leq 2 \cos \theta\), the bounds for \(r\) are set. For \(z\), the height of the cylinder is 4, so \(z\) ranges from 0 to 4. The cylinder is rotationally symmetric around the z-axis, so \(\theta\) ranges from \(0\) to \(2\pi\). With these bounds, we can set up the integral for the volume as: $$V = \int_{0}^{2 \pi} \int_{0}^{4} \int_{0}^{2 \cos \theta} r \, dr \, dz \, d\theta$$

Step 3: Evaluate the integral

Now, we evaluate the integral step by step. First, integrate with respect to \(r\): $$V = \int_{0}^{2 \pi} \int_{0}^{4} \left[\frac{1}{2} r^2\right]_{0}^{2 \cos \theta} dz \, d\theta = \int_{0}^{2 \pi} \int_{0}^{4} 2 \cos^2 \theta \, dz \, d\theta$$ Next, integrate with respect to \(z\): $$V = \int_{0}^{2 \pi} \left[2z \cos^2 \theta\right]_{0}^{4} d\theta = \int_{0}^{2 \pi} 8 \cos^2 \theta \, d\theta $$ Finally, integrate with respect to \(\theta\): $$V = \int_{0}^{2 \pi} 8 \cos^2 \theta \, d\theta$$ To evaluate this integral, we can use the double-angle formula for cosine: \(\cos{2 \theta} = 2 \cos^2 \theta - 1\). Solving for \(\cos^2 \theta\), we have \(\cos^2 \theta = \frac{1}{2}(\cos{2\theta} + 1)\). Now, substitute this expression back into the integral and evaluate: $$V = 8 \int_{0}^{2 \pi} \frac{1}{2}(\cos{2 \theta} + 1) \, d\theta = 4\left[\frac{1}{2}\sin{2\theta} + \theta\right]_{0}^{2 \pi}$$ Calculate the final result: $$V = 4\left[\frac{1}{2}(\sin{4 \pi} - \sin0) + (2 \pi - 0)\right] = 4(0 + 2 \pi) = 8 \pi$$

Step 4: Conclusion

The volume of the solid cylinder with height 4 and base disk \(0 \leq r \leq 2 \cos \theta\) in cylindrical coordinates is \(V = 8 \pi\).

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on determining the total size, often referred to as the 'integral,' of quantities like areas under curves, volumes of solids, and much more. In the context of finding the volume of solids, the integral can be visualized as an accumulation of infinitesimals—tiny slices of the solid that when summed up give the entire volume.

Integral calculus provides the tools necessary for solving complex geometrical problems through its two fundamental concepts: indefinite and definite integrals. The indefinite integral represents a family of functions (antiderivatives), while the definite integral gives a numerical solution corresponding to the total accumulation of the quantity of interest over a particular interval. When calculating volumes, the definite integral is the main focus, as it allows us to quantify the space occupied by a three-dimensional object.

In the exercise provided, the volume of the cylindrical solid is found by setting up and evaluating a triple integral. Each integral represents accumulation in one dimension of the cylindrical coordinate system. By solving this step by step, we effectively sum up the contributions of each volume element to the total volume of the solid, adhering to the specific limits of integration.

Volume of Solids

Understanding the volume of solids is a key application of integral calculus. A solid's volume is a measure of how much three-dimensional space it occupies, and calculating this value often requires integrating over the object's domain. Different shapes and topologies require different approaches and, at times, coordinate systems.

For simpler objects like cubes or prisms, we can often calculate volume through geometric formulas, such as multiplying length by width by height. However, for more complex shapes, we turn to integral calculus. By breaking a solid down into infinitesimally small pieces, we calculate the volume of each piece and then use integration to add them all together to find the solid’s total volume.

In our case, the solid is a cylinder, and the integral calculus method involves setting up a triple integral. Each level of integration accounts for a different dimension within the cylinder. By using the specific volume element in cylindrical coordinates, we can integrate over the radius, angle, and height to find this particular cylinder's volume.

Cylindrical Coordinate System

The cylindrical coordinate system is an extension of polar coordinates into three dimensions, which is particularly effective for dealing with problems that exhibit rotational symmetry, such as cylinders and cones. A point in this system is represented by \( (r, \theta, z) \), where \( r \) is the radial distance from the z-axis, \( \theta \) is the azimuthal angle from the positive x-axis, and \( z \) is the height above the xy-plane.

In cylindrical coordinates, the volume element is given by \( dV = r \, dr \, d\theta \, dz \). This element represents a small volume within the cylinder that is theoretically infinitesimal, and it's crucial for calculating the volume of solids. The presence of the radial term, \( r \), recognizes that as the radius increases, the circumference—and hence the volume of a thin slice—also increases.

When solving volume problems in cylindrical coordinates, as in the provided exercise, we always identify the bounds for each component (\( r \) and \( \theta \) and \( z \)) based on the geometry of the problem. Then, we integrate the volume element across these bounds to find the total volume. The evaluation of the integral carefully follows the order of \( r \), \( \theta \), and \( z \), acknowledging the nested nature of these components in the context of the object’s geometry.