Problem 35 - Thomas Calculus

Question

Give the limits of integration for evaluating the integral $$ \iiint f(r, \theta, z) d z r d r d \theta $$ as an iterated integral over the region that is bounded below by the plane \(z=0,\) on the side by the cylinder \(r=\cos \theta,\) and on top by the paraboloid \(z=3 r^{2}\) .

Step-by-Step Solution

Answer

The limits are \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\), \(0 \leq r \leq \cos \theta\), \(0 \leq z \leq 3r^2\).

Step 1: Identify the Boundaries for r

The region is bounded on the side by the cylinder defined by the equation \( r = \cos \theta \). This restricts the radial coordinate \( r \) from 0 to \( \cos \theta \). We'll conclude the range for \( r \) is given by \( 0 \leq r \leq \cos \theta \).

Step 2: Identify the Boundaries for z

The problem states the region is bounded below by the plane \( z = 0 \) and on top by the paraboloid \( z = 3r^2 \). Therefore, the limits for \( z \) are from \( 0 \) to \( 3r^2 \).

Step 3: Determine the Range for θ

For the cylinder \( r = \cos \theta \), \( \theta \) varies over the interval where the cylindrical surface is consistent. Since \( \cos \theta \) is non-negative, this interval is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Step 4: Combine the Limits into an Iterated Integral

The given triple integral expression can now be rewritten with the determined limits: \[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_{0}^{\cos \theta} \int_{0}^{3r^2} f(r, \theta, z) \cdot dz \cdot r \cdot dr \cdot d\theta.\]

Key Concepts

Integration Limits

Integration limits are crucial when setting up a triple integral because they define the region over which the function is being integrated. In our case, we are working with a spatial region bounded by specific geometric shapes, which guides the determination of these limits.

• For the radial coordinate, \( r \), the limit is defined by the cylinder \( r = \cos \theta \). This means that for each \( \theta \), \( r \) ranges from 0 up to \( \cos \theta \).
• The vertical coordinate, \( z \), is constrained by a plane at \( z=0 \) below and by a paraboloid \( z=3r^2 \) above. Therefore, \( z \) ranges from 0 to \( 3r^2 \).
• Lastly, the angular coordinate, \( \theta \), is governed by the symmetry and nature of the cosine function. The range is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) because \( r = \cos \theta \) must remain non-negative.

Setting these limits correctly is essential in correctly defining the volume or area of integration.

Cylindrical Coordinates

Cylindrical coordinates are an extension of polar coordinates to three dimensions and they are useful for solving problems with symmetry about an axis. These coordinates consist of \( (r, \theta, z) \), where:

• \( r \) is the radial distance from the z-axis.
• \( \theta \) is the angle in the xy-plane from the positive x-axis.
• \( z \) represents the height along the z-axis and is analogous to conventional Cartesian z.

The beauty of cylindrical coordinates is that they can greatly simplify the evaluation of the triple integrals, especially when the boundaries of the region of integration have cylindrical symmetry. In this problem, using these coordinates helps to naturally express the boundaries defined by a cylinder and paraboloid.

Paraboloid

A paraboloid is a three-dimensional surface that can be thought of as a parabola extended along an axis. In our problem, the top boundary of the integration region is described by the equation \( z = 3r^2 \). This represents a paraboloid that opens upwards along the z-axis.

• The surface flares outward more quickly as \( r \) increases because \( z \) is proportional to the square of \( r \).
• For any fixed \( \theta \), as \( r \) increases from 0 to its maximum value of \( \cos \theta \), \( z \) will increase from 0 up to \( 3 \cos^2 \theta \).

The parabolic nature defines how high we are integrating in the z-direction at any point in the \((r, \theta)\) plane.

Cylinder

A cylinder in this context is a three-dimensional surface of constant radial distance from an axis. The equation \( r = \cos \theta \) describes the bounds on one side of our region of integration.

• This equation delineates a surface where the radial distance varies depending on \( \theta \), peaking when \( \theta = 0 \) at \( r = 1 \).
• The cylinder effectively cuts off the portion of the paraboloid that lies beyond this radial limit, restricting the maximum value of \( r \).
• Since we are dealing with these boundaries in a \(r-\theta\) framework, this expresses a condition that the region does not extend beyond the intersection points mapped out by the cylindrical surface.

Understanding the role of this boundary is essential to visualize how the region of integration is formed and how the defined limits come into play.