Question

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The wedge cut from the cardioid cylinder \(r=1+\cos \theta\) by the planes \(z=2-x\) and \(z=x-2\)

Short answer

Answer: The volume of the wedge cut from the cardioid cylinder is 0 cubic units, indicating that the planes do not intersect the cardioid.

Step-by-step solution

Determine the limits of integration for r, θ, and z

To find the limits of integration for r, we already know that the cylinder is defined by \(r=1+\cos \theta\). By examining the cardioid, we can see that the range for r is from 0 to \(2\cos\theta\). For θ, we need to determine the full extent of the cardioid which is from 0 to \(2\pi\). For z, the solid is bound by the planes \(z=2-x\) and \(z=x-2\). In the cylindrical coordinate system, we can write these equations as: - \(z = 2 - r \cos \theta\) - \(z = r \cos \theta - 2\) Now we have the limits of integration for r, θ, and z.

Set up the volume integration

With the limits of integration determined, we can set up the triple integral to find the volume of the solid: \(V = \int_{0}^{2\pi} \int_{0}^{2\cos\theta} \int_{r\cos\theta-2}^{2-r\cos\theta} r \, dz \, dr \, d\theta\)

Integrate with respect to z

First, we integrate with respect to z: \(V = \int_{0}^{2\pi} \int_{0}^{2\cos\theta} r[(2-r\cos\theta) - (r\cos\theta-2)] \, dr \, d\theta\) This simplifies to: \(V = \int_{0}^{2\pi} \int_{0}^{2\cos\theta} 2r - 2r^2\cos\theta \, dr \, d\theta\)

Integrate with respect to r

Now, we integrate with respect to r: \(V = \int_{0}^{2\pi} [ (r^2 - \frac{2}{3}r^3\cos\theta) \Big|_0^{2\cos\theta} ] \, d\theta\) This evaluates to: \(V = \int_{0}^{2\pi} (4\cos\theta - \frac{16}{3}\cos^3\theta) \, d\theta\)

Integrate with respect to θ

Finally, we integrate with respect to θ: \(V = \left[ 4\sin\theta - \frac{4}{3}\sin^3\theta \right]_0^{2\pi}\) Since \(\sin (0) = 0\) and \(\sin (2\pi) = 0\), the volume becomes: \(V = 0 - 0 = 0\) In this case, the volume of the wedge cut from the cardioid cylinder is zero, which indicates that the planes do not intersect the cardioid.

Key concepts

Triple Integral

A triple integral extends the concept of an integral to functions of three variables. In the world of mathematics and physics, it allows us to calculate volumes and mass of three-dimensional regions, among other applications. The triple integral is written as \( \int \int \int f(x,y,z) \, dx \, dy \, dz \), where \( f(x,y,z) \) is the function being integrated over a three-dimensional region.
For calculating the volume of a solid using a triple integral, you consider the value of \( f(x,y,z) \) as 1, effectively 'summing up' tiny blocks of space to find the total volume. In the exercise, the triple integral was set up using appropriate limits of integration to solve for the volume of a wedge cut from a cardioid cylinder.

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system where each point is defined by a radius (r), an angle (\(\theta\)), and a height (z). These are incredibly useful when dealing with symmetrical figures around a central axis, like cylinders or cones.
The cardioid cylinder problem is well-suited to cylindrical coordinates, as the radial symmetry of the shape lends to simpler equations and integrals in this system. The cardioid equation given in the exercise \( r=1+\cos \theta \) describes the radius of a point on the cylinder as a function of the angle \(\theta\), further supporting the choice of this coordinate system for the problem at hand.

Limits of Integration

The limits of integration are the bounds within which we are integrating. In one-dimensional integrals, these are simply two values on the number line. For triple integrals, these limits become more intricate, as we're working within a three-dimensional space and need to set bounds for each variable: r, \(\theta\), and z.
Selecting the correct limits of integration is critical for the solution, as these define the specific region in space we're interested in. In our cardioid cylinder example, determining the range for r and \(\theta\) required analyzing the shape's geometry, while the limits for z were found by re-expressing the given plane equations in cylindrical coordinates.
Upon setting these limits appropriately, the integral calculation becomes manageable, with the limits steering the direction of integration over the region of interest.

Volume Calculation

Volume calculation using triple integrals is a powerful technique in multivariable calculus. After setting up an integral with the correct function and limits of integration, the process involves integrating over the variables one at a time. Each integration step takes us closer to the final volume of the solid.
In the provided example, we were calculating the volume of a wedge cut from the cardioid cylinder. The initial integral setup took into consideration the geometry of the cylinder and the planes cutting through it. Subsequent integration steps simplified the problem down to a final integral, which, after evaluation, surprisingly resulted in zero. This indicates that no volume is enclosed between the planes and the cardioid, which may suggest a possible miscalculation or that the planes do not intersect the region defined by the cardioid cylinder, as pointed out in the solution.