Question

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of the cap of a sphere of radius \(R\) with thickness \(h\).

Short answer

Question: Determine the volume of the cap of a sphere with radius \(R\) and thickness \(h\). Answer: The volume of the cap of the sphere is \(V = \frac{2\pi}{3}\left(R^3 - (R-h)^3\right)\).

Step-by-step solution

Determine the equation of bounding surfaces and the cap thickness

We start by considering a sphere of radius \(R\) with a hemisphere cap formed by slicing a smaller sphere of radius \(r\) out of the main sphere. The thickness of the cap is the difference between the radii of the two spheres, which we denote as \(h\), such that \(h = R - r\). Therefore, the sphere with radius \(r\) has the following equation in spherical coordinates: $$r(\theta, \phi) = R - h.$$

Specify the limits of integration

We will now define the limits of integration in spherical coordinates. For the angle \(\theta\), we need to consider the entire hemisphere, so \(\theta\) ranges from 0 to \(\pi/2\): $$0 \le \theta \le \frac{\pi}{2}.$$ For the angle \(\phi\), we need to take into account the full 360 degrees of the cap, so \(\phi\) ranges from 0 to \(2\pi\): $$0 \le \phi \le 2\pi.$$ Lastly, the radial distance \(r\) starts from \(R - h\) (the smaller sphere) to \(R\) (the larger sphere): $$R - h \le r \le R.$$

Set up the triple integral

Using the volume element in spherical coordinates, \(dV = r^2 \sin\theta \, dr \, d\theta \, d\phi\), we can define the triple integral representing the volume of the cap: $$V = \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{R-h}^{R} r^2 \sin\theta \, dr \, d\theta \, d\phi.$$

Evaluate the triple integral

We can now evaluate the integral one variable at a time, starting with \(r\): $$V = \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \left[ \frac{1}{3} r^3 \right]_{R-h}^{R} \sin\theta \, d\theta \, d\phi = \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} (\frac{1}{3}R^3 - \frac{1}{3}(R - h)^3) \sin\theta \, d\theta \, d\phi.$$ Next, we integrate with respect to \(\theta\): $$V = \int_{0}^{2\pi} \left[ -\frac{1}{3}R^3 \cos\theta + \frac{1}{3}(R-h)^3 \cos\theta \right]_{0}^{\frac{\pi}{2}} d\phi = \int_{0}^{2\pi}\left(\frac{1}{3}R^3 - \frac{1}{3}(R-h)^3\right) d\phi.$$ Finally, integrating with respect to \(\phi\): $$V = \left[ \frac{1}{3}\phi\left(R^3 - (R-h)^3\right)\right]_{0}^{2\pi} = \frac{2\pi}{3}\left(R^3 - (R-h)^3\right).$$ So, the volume of the cap of the sphere is: $$V = \frac{2\pi}{3}\left(R^3 - (R-h)^3\right).$$

Key concepts

Triple Integration

Triple integration is a method used to calculate the volume of a solid by integrating over three variables. It expands upon the concept of double integration, which finds areas, by adding a third dimension for volume.
This process involves calculating an integral over a three-dimensional region, often using a volume element such as \(dV = dx \times dy \times dz\) or in spherical coordinates, \(dV = r^2 \sin\theta \, dr \, d\theta \, d\phi\).

Steps to set up a triple integral include:

• Identifying the region of integration.
• Choosing the correct coordinate system.
• Setting up the limits for each variable.
• Integrating in the prescribed sequence.

Use triple integration to efficiently solve problems involving complex 3D shapes.

Spherical Coordinates

Spherical coordinates are used to simplify integration when dealing with spherical or circular shapes. In these coordinates, a point in space is defined by:

• \( r \): the radial distance from the origin to the point.
• \( \theta \): the polar angle in the \(xy\)-plane from the positive \(x\)-axis.
• \( \phi \): the azimuthal angle from the positive \(z\)-axis.

This system is especially useful when finding volumes of objects like spheres or cones.

The conversion between Cartesian and spherical coordinates helps in setting up the integral appropriately, often leading to simpler limits and integrands. This coordinate transformation is characterized by:
\( x = r \sin\theta \cos\phi \), \( y = r \sin\theta \sin\phi \), \( z = r \cos\theta \).

Bounding Surfaces

Bounding surfaces define the limits and shape of the region you are integrating over.
For the cap of a sphere, we consider surfaces formed by intersecting spheres.

In our example, the sphere of radius \(R\) and the sphere of radius \(r = R-h\) create a cap when intersected. These surfaces determine the radial limits in spherical coordinates:

• The outer surface is the sphere \( r = R \).
• The inner boundary is the sphere \( r = R-h \).

Understanding bounding surfaces enables you to set accurate regions for integration, essential in capturing the entire solid's volume.

Limits of Integration

Setting the correct limits of integration is crucial in solving triple integrals. These limits define the range over which each variable is integrated.
In our case of finding the volume of a spherical cap:

• \( \theta \) ranges from \( 0 \) to \( \frac{\pi}{2} \) to capture the hemisphere.
• \( \phi \) varies from \( 0 \) to \( 2\pi \) for full rotation.
• Radial distance \( r \) runs from \( R-h \) to \( R \).

Precise limits ensure the volume calculated is precise, capturing only the intended portion of the sphere. Being careful with limits helps avoid overestimating or underestimating the volume.