Problem 15 - Thomas Calculus

Question

Set up the iterated integral for evaluating \(\iiint_{D} f(r, \theta, z) d z r d r d \theta\) over the given region \(D .\) \(D\) is the right circular cylinder whose is the circle \(r=2 \sin \theta\) in the \(x y\) -plane and whose top lies in the plane \(z=4-y\)

Step-by-Step Solution

Answer

The iterated integral is \( \int_0^{\pi} \int_0^{2 \sin \theta} \int_0^{4 - r \sin \theta} f(r, \theta, z) \cdot r \; dz \; dr \; d\theta. \)

Step 1: Analyze the given region D

The region \(D\) is a right circular cylinder. The base of the cylinder is a circle in the \(xy\)-plane given by \(r = 2 \sin \theta\), which represents our radial boundaries. The top of the cylinder is defined by a plane \(z = 4 - y\).

Step 2: Establish boundaries for r and θ

In polar coordinates, the equation \(r = 2 \sin \theta\) represents a circle. The value of \(r\) varies from 0 to \(2 \sin \theta\). Since \(2 \sin \theta\) can range from 0 to 2 (the maximum sine value), \(\theta\) ranges from 0 to \(\pi\).

Step 3: Set the boundary for z

The boundary for \(z\) is given by the plane equation \(z = 4 - y\). In polar coordinates, \(y = r \sin \theta\), so \(z\) ranges from 0 to \(4 - r \sin \theta\).

Step 4: Set up the iterated integral

The iterated integral over the region \(D\) in cylindrical coordinates is set as follows: \[ \int_0^{\pi} \int_0^{2 \sin \theta} \int_0^{4 - r \sin \theta} f(r, \theta, z) \cdot r \; dz \; dr \; d\theta. \]

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a system of coordinates used for the representation of points in a three-dimensional space. This system is an extension of polar coordinates, which are used in two dimensions. The cylindrical coordinates of a point are expressed as

• \( r \) - the radial distance from the origin to the point's projection in the \( xy \)-plane.
• \( \theta \) - the angular coordinate, which is the angle measured from the positive \( x \)-axis to the line connecting the projection of the point to the origin.
• \( z \) - the height above the \( xy \)-plane.

For problems involving regions like cylinders, this coordinate system simplifies calculations greatly. The given exercise involves finding the region's boundaries and setting up an iterated integral within the cylindrical coordinate framework. This setup automatically accounts for circular symmetry in three-dimensional problems, making integration easier and more intuitive.

Polar Coordinates

Polar coordinates serve as a two-dimensional system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This system is especially useful when dealing with problems featuring symmetry about an origin, like circles and spirals. An important aspect of polar coordinates is:

• \( r \) - representing the radius, or the straight-line distance from the origin to the point.
• \( \theta \) - showing the angle formed with the positive x-axis.

In the exercise, the boundary of the region is given by \( r = 2 \sin \theta \), describing a circle in polar terms. This expression defines how \( r \) changes as a function of \( \theta \). Since \( \theta \) varies from 0 to \( \pi \), the setup allows capturing the geometry of half a circle. In conjunction with cylindrical coordinates, polar coordinates help in efficiently solving volumes enclosed by symmetrical boundaries.

Triple Integrals

Triple integrals are a way to extend the concept of integration to three dimensions. They allow for the calculation of volumes under surfaces in three-dimensional space. When evaluating a triple integral, we integrate a function over a three-dimensional region, iteratively performing integration with respect to each coordinate. This is reflected in the order of integration:

• First integrating with respect to \( z \), the height, from 0 to \( 4 - r \sin \theta \).
• Then with respect to \( r \), from 0 to the function \( 2 \sin \theta \).
• Finally with respect to \( \theta \), from 0 to \( \pi \).

This order aligns with the nature of the problem, moving successively from vertical boundaries to radial boundaries, concluding with angular boundaries. The setup and evaluation of triple integrals are key to understanding many physical phenomena and geometrical calculations.

Integration Boundaries

Integration boundaries define the limits over which integration takes place. In three dimensions, for iterated integrals like triple integrals, it's essential to correctly define these limits for each coordinate. This ensures the integral correctly represents the volume or the space within the given region.

• The \( z \)-boundary in our integration is defined, as per the exercise, by the plane \( z = 4 - y \). Converting \( y \) to cylindrical coordinates gives \( z = 4 - r \sin \theta \), maintaining consistent variable use.
• For \( r \), the integration limits vary from 0 to \( 2 \sin \theta \), which are defined via the circle in the \( xy \)-plane given by the equation \( r = 2 \sin \theta \).
• Lastly, \( \theta \) ranges from 0 to \( \pi \), encompassing half the circle described.

Setting these boundaries correctly is paramount to solving the integral and finding the volume or other related measurements accurately. Each boundary corresponds to a physical or geometrical aspect of the problem, translating directly into the limits applied in integration.