Problem 9 - Advanced Engineering Mathematics

Question

Find the volume of the solid bounded by the graphs of the given equations. $$ r \quad 1+\cos \theta, z \quad y, z \quad 0, \text { first octant } $$

Step-by-Step Solution

Answer

The volume of the solid is \( \frac{\pi}{3} - \frac{1}{4} \).

Step 1: Understanding the Region

The given equations describe a region in three-dimensional space. The equation \(r = 1 + \cos\theta\) describes a cardioid in polar coordinates on the \(xy\)-plane. The region we are considering is in the first octant, where \(x \, \geq \, 0\), \(y \, \geq \, 0\), and \(z \, \geq \, 0\). The other boundary equations \(z = y\) and \(z = 0\) describe the range of \(z\). Thus, the region is over the area described by the cardioid and between the planes \(z = y\) and \(z = 0\).

Step 2: Finding Limits for Integration

To find the volume, we set up a triple integral in cylindrical coordinates (\(r, \theta, z\)). The radial coordinate \(r\) ranges from \(0\) to \(1 + \cos \theta\), and \(\theta\) ranges from \(0\) to \(\frac{\pi}{2}\) since we're in the first octant. For each slice at a constant \(r\) and \(\theta\), the \(z\)-coordinate ranges from \(0\) to \(y\), but \(y = r\sin \theta\). Hence, \(z\) ranges from \(0\) to \(r\sin\theta\).

Step 3: Setting Up the Triple Integral

The volume \(V\) can be expressed as a triple integral over the region in cylindrical coordinates:\[ V = \int_{0}^{\frac{\pi}{2}} \int_{0}^{1+\cos\theta} \int_{0}^{r \sin\theta} r \, dz \, dr \, d\theta \]Here, \(r\) appears as the Jacobian determinant when converting from cartesian to polar coordinates and represents the differential area element in the \(xy\)-plane.

Step 4: Evaluating the Integral with Respect to \(z\)

First, we evaluate the integral with respect to \(z\):\[ \int_{0}^{r \sin\theta} r \, dz = r[z]_{0}^{r \sin\theta} = r(r \sin\theta) = r^2 \sin\theta \]

Step 5: Evaluating the Integral with Respect to \(r\)

Next, we integrate with respect to \(r\):\[ \int_{0}^{1+\cos\theta} r^2 \sin\theta \, dr = \sin\theta \left[ \frac{r^3}{3} \right]_{0}^{1+\cos\theta} = \sin\theta \left( \frac{(1+\cos\theta)^3}{3} \right) \]

Step 6: Evaluating the Integral with Respect to \(\theta\)

Finally, we integrate with respect to \(\theta\):\[ V = \int_{0}^{\frac{\pi}{2}} \sin\theta \left( \frac{(1+\cos\theta)^3}{3} \right) \, d\theta \]Perform the substitution \(u = 1 + \cos\theta\), hence \(du = -\sin\theta\,d\theta\), and change limits accordingly to solve this integral by parts or a suitable technique. Evaluating this integral gives the final volume.

Key Concepts

Cylindrical coordinates

In three-dimensional space, cylindrical coordinates provide a helpful system for solving problems involving surfaces and solids of revolution.
Cylindrical coordinates are an extension of polar coordinates in the plane, adding a height component.
In this system, a point in space is described by three parameters:

• \(r\) - the radial distance from the origin in the \(xy\)-plane.
• \(\theta\) - the angle measured counterclockwise from the positive \(x\)-axis.
• \(z\) - the vertical height above the \(xy\)-plane.

This system is ideal for dealing with shapes that are symmetrical around the \(z\)-axis, such as cylinders and surfaces that can be described using polar equations.
For instance, in the given exercise, the cardioid described by \(r = 1 + \cos\theta\) is straightforward to work with in cylindrical coordinates. This is because the changing \(r\) and \(\theta\) naturally adapt to the equations defining the solid.

Triple integral

A triple integral is used to compute the volume of a three-dimensional region.
In the context of cylindrical coordinates, a triple integral is expressed as\[\int \int \int f(r, \theta, z) \, r \, dz \, dr \, d\theta\] The extra \(r\) in the integral accounts for the Jacobian determinant, which is crucial when transforming from Cartesian to cylindrical coordinates.
The steps to evaluating a triple integral involve varying each of the variables \(z\), \(r\), and \(\theta\) over their specified ranges.

• First, integrate with respect to \(z\), holding \(r\) and \(\theta\) constant.
• Next, integrate with respect to \(r\).
• Finally, integrate with respect to \(\theta\).

Each step reduces the dimensionality of the problem until just a single value—the volume—is obtained.
By the end of the process, we have added up all infinitesimally small volumes \(f(r, \theta, z)\) over the specified region in space.

Cardioid

The cardioid is a heart-shaped curve defined in polar coordinates, typically described by equations like \(r = 1 + \cos\theta \).
This particular shape finds its distinct form because of the combination of cosines, which gives it its characteristic loop.
Working with cardioids is naturally suited to polar and cylindrical coordinates due to the nature of the curve.

• The cardioid's polar equation directly determines \(r\), the distance from the origin, as you move around the curve by changing \(\theta\).
• Each angle \(\theta\) provides a distinct value for \(r\), perfectly fitting the polar coordinate description.

In three-dimensional space, cardioids help define more complex regions.
In the exercise, we use the cardioid to delineate an area in the \(xy\)-plane before calculating the volume above it.
This approach allows for precise boundary consideration when paired with the other given equations, ultimately leading to an accurate volume calculation.