Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the area of the part of the surface of a sphere of radius \(a\) between two parallel planes \(h\) units apart \((h<2 a)\) is \(2 \pi a h .\) Thus, show that if a right circular cylinder is circumscribed about a sphere then two planes parallel to the base of the cylinder bound regions of the same area on the sphere and the cylinder.

Step-by-Step Solution

Answer

The area on the sphere between parallel planes is \(2\pi a h\), matching the same region on the circumscribed cylinder.

Step 1: Understand the Sphere's Geometry

Consider a sphere of radius \(a\). The area of its surface is given by the formula \[ 4 \pi a^2. \] We are interested in the area of the surface portion between two parallel planes that are \(h\) units apart.

Step 2: Identify the Band Area on the Sphere

Because the sphere is symmetric, the region between two planes forms a band around the sphere. This band has constant width \(h\) on the surface of the sphere.

Step 3: Use the Surface Integral for the Band Area

The area of the band on the sphere can be calculated using the surface integral of a curved surface. For a band with height \(h\), centered at the equator of the sphere, the resulting area from integral calculations is found to be \[ 2 \pi a h. \]

Step 4: Comparing with the Cylinder

Consider a right circular cylinder circumscribed around the sphere, with its height equal to the diameter of the sphere \((2a)\). The lateral surface of this cylinder is \[ 2 \pi a \cdot 2a = 4 \pi a^2. \]

Step 5: Analyze the Region Bounded by Two Planes

For both the sphere and the circumscribed cylinder, if two planes parallel to the base of the cylinder cut the sphere, the band formed on the surface of the sphere will have an area of \(2 \pi a h\), identical to the band on the cylinder's lateral surface in this bounded region.

Key Concepts

Geometry of Spheres

Spheres are unique 3-dimensional shapes where all points on the surface are equidistant from a central point known as the center of the sphere. This distance from the center to any point on the surface is referred to as the radius \(a\). The spherical shape is perfectly symmetrical across any axis that passes through the center, making it a fundamental object in geometry.
Spheres have several important properties:

• The total surface area of a sphere is calculated using the formula \(4 \pi a^2\), where \(a\) is the radius.
• A sphere's symmetry is leveraged when calculating specific areas, such as segments or bands formed by intersecting planes.
• Due to its symmetry, any plane that cuts the sphere forms a circular boundary on its surface, creating a surface segment or a spherical cap.

When two parallel planes intersect the sphere, they create a band, or zonal area, between them. If these planes are \(h\) units apart, the band can be visualized as a belt-like strip wrapping around the sphere, helping to understand how sphere geometry plays a role in calculating specific areas.

Cylindrical Surfaces

Cylindrical surfaces emerge from the rotation of a line parallel to an axis, creating a round solid known as a cylinder. Understanding how cylindrical surfaces relate to spheres involves looking at special arrangements like the circumscribed cylinder scenario.
A right circular cylinder—a common cylinder in geometry—aligns perfectly with the sphere when it is circumscribed around it:

• This particular arrangement means the cylinder's height equals the diameter of the sphere \(2a\).
• The lateral surface area of the cylinder is given by the equation \(2 \pi a \times 2a = 4 \pi a^2\), equal to the sphere’s total surface area.
• Thus, the symmetry between the sphere and cylinder helps simplify area comparisons, crucial when discussing bounded areas.
  

In geometry exercises like this one, it's insightful to see how a simple shape like a cylinder can help compare areas and develop an understanding of more complex surface properties.

Surface Integrals

Surface integrals are powerful mathematical tools used to find areas on curved surfaces like spheres. They involve integrating over a surface as opposed to a simple line or volume. In this context, surface integrals allow us to calculate areas of surface segments or bands on spheres, like the one formed between two parallel planes.
Surface integrals help accomplish several tasks:

• They provide the area of the band \(2 \pi a h\) on the sphere, affirming the area found by symmetry and theoretical analysis.
• During integration, symmetry of the sphere aids in simplifying the integral setup and evaluation.
• Enables precise calculation for areas bound by specific constraints such as those set by cutting planes.

Practically speaking, utilizing surface integrals in exercises like this solidifies understanding of how integral calculus can directly apply to geometric problems, reinforcing concepts of symmetry and area computation.