Problem 79 - Thomas Calculus

Question

Cylinder and planes Find the volume of the region enclosed by the cylinder \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(y+z=4\) .

Step-by-Step Solution

Answer

The volume is \(16\pi\).

Step 1: Analyze the Cylinder

The equation of the cylinder is given by \(x^2 + y^2 = 4\). This indicates a circular cylinder with radius 2, centered along the z-axis.

Step 2: Determine the Bounding Planes

The plane \(z = 0\) corresponds to the base of the region of interest. The plane \(y + z = 4\) can be rearranged to \(z = 4 - y\), indicating a plane tilted with respect to the y-axis.

Step 3: Find the Limits of Integration

For the given cylinder, the integration limits for \(x\) will be \(-2\) to \(2\) and for \(y\), it will be from \(\sqrt{4 - x^2}\) to \(-\sqrt{4 - x^2}\). The region where the plane \(z = 4 - y\) intersects the cylinder must also be considered for the z-limits, i.e., from \(0\) to \(4 - y\).

Step 4: Set Up the Integral

The volume \(V\) of the region can be expressed as a triple integral in cylindrical coordinates. Convert the coordinates using \(x = r\cos\theta\), \(y = r\sin\theta\), and the Jacobian \(r\,dr\,d\theta\). This gives:\[ V = \int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{4 - r\sin\theta} r\,dz\,dr\,d\theta \]

Step 5: Evaluate the Integral

First evaluate the inner integral with respect to \(z\):\[ \int_{0}^{4 - r\sin\theta} r\,dz = r(4 - r\sin\theta) \]Now substitute back and compute the outer integrals:\[ V = \int_{0}^{2\pi} \int_{0}^{2} r(4 - r\sin\theta)\,dr\,d\theta \]\[ V = \int_{0}^{2\pi} \left[4r^2/2 - r^3\sin\theta /3\right]_{0}^{2}\,d\theta \]\[ V = \int_{0}^{2\pi} \left[8 - \frac{8\sin\theta}{3}\right] \,d\theta \]

Step 6: Final Integration and Solution

Evaluate the remaining integral:\[ V = \left[8\theta - \frac{8}{3}(-\cos\theta)\right]_{0}^{2\pi} \]This simplifies to:\[ V = 8(2\pi) - \frac{8}{3}(\cos(2\pi) - \cos(0)) = 16\pi \] (since \(\cos(2\pi) = \cos(0) = 1\))

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates offer a useful way to represent points in 3D space, especially when dealing with problems involving symmetry around a central axis. These coordinates consist of three components: the radial distance \( r \), the angle \( \theta \), and the height \( z \).
This system is especially beneficial when working with cylinders or other similar structures, as it allows for straightforward integration that aligns neatly with circular shapes.
Cylindrical coordinates transform the usual Cartesian coordinates \((x, y, z)\) into:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( z = z \)

In many problems, such as finding volumes of solids, the Jacobian determinant in these coordinates, \( r \,dr \,d\theta \), plays a critical role. This accounts for the transformation from rectangular grid-like cells to circular slices, making integration feasible and accurate.

Triple Integrals

Triple integrals are extended integrations used to compute volumes and other 3D quantities by accumulating the effect of a smaller piece (like area) over a specified region.

When dealing with geometric volumes, such as the one formed by a cylinder capped by planes, triple integrals can be set in either Cartesian or cylindrical coordinates, the latter is often simpler for shapes with circular symmetry.

Using cylindrical coordinates, the triple integral setup follows a certain sequence:

• First integral (innermost) with respect to \( z \): accounts for the height or depth of the slice.


• Second integral with respect to \( r \): captures radial dimensions, moving outward from the central axis.


• Third integral with respect to \( \theta \): completes the full sweep around the axis.

The calculations proceed step-by-step, solving from the inner to the outer integral. Each integration refines our understanding of the volume under analysis.

Cylinder and Planes

In this problem, understanding the way a geometric cylinder interacts with planes is essential. The equation \(x^{2} + y^{2} = 4\) describes a vertical cylinder with a radius of 2, centered on the z-axis.
This cylinder's surface is uniform along its height, forming a circular cross-section when viewed from above.

The planes bounding this cylinder are:

• \( z = 0 \): This plane is flat, setting the base boundary for the region of interest at this level.


• \( y + z = 4 \): Transforms to \( z = 4 - y \), creating a tilted plane that 'caps' the cylinder, sloping down as \( y \) increases.

It's crucial to visualize the interaction between these structures: The plane intersects the cylinder and forms a slanted top surface, above its base \( z = 0 \). The challenge is to explore how these surfaces enclose a volume and limit integration calculations.

Limits of Integration

When calculating a volume using integrals, setting the correct limits of integration is a vital step.
These limits define the boundaries of integration, ensuring that only the region of interest is considered.

For the cylindrical coordinates approach:

• \( r \): ranges from 0 to 2, as it spans the radius of the cylinder.


• \( \theta \): is from 0 to \( 2\pi \), capturing the entire circular path around the cylinder.


• \( z \): ranges from 0 to \( 4 - r\sin\theta \), reflecting the volume constrained beneath the slanted plane \( y+z = 4 \).

These limits ensure all calculations remain within the designated region, fully capturing the volume created by these intersecting geometries without any gaps or overreaches.