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 of a spherical cap is \(2 \pi a h\), matching bounded cylindrical regions.

Step 1: Understand the Geometry

We have a sphere of radius \(a\) with two parallel planes cutting through it, and these planes are \(h\) units apart. We are to find the area of the spherical surface sandwiched between these planes. This essentially forms a spherical cap with height \(h\).

Step 2: Area of Spherical Cap Formula

The formula for the surface area of a spherical cap of height \(h\) from a sphere of radius \(a\) is \(2 \pi a h\). This result comes from using calculus and geometry, considering the lateral surface area that forms on the curved surface between the two planes.

Step 3: Derive Area of Spherical Cap

For a more rigorous derivation: consider slicing the sphere with equally spaced planes. Note that each slice forms a ring with area \(2 \pi a \Delta y\), where \(\Delta y\) is an infinitesimally small change in the height of the "cap." Integrating these from 0 to \(h\) yields the full formula's surface area \(2 \pi a h\).

Step 4: Compare with Cylinder Area

Now consider a right circular cylinder circumscribed about a sphere of radius \(a\). The cylinder’s height is \(2a\), the diameter. When two parallel planes are drawn at the sphere's equator, the cylinder's curved surface area is \(2 \pi a (2a) = 4 \pi a^2\), which is shared equally top and bottom, bounding regions \(2 \pi a h\) each.

Step 5: Regions of Equal Area

Hence, when two planes are parallel and bound a spherical cap as well as a cylindrical section, they have the same surface areas, as deduced from the same height \(h\) above the sphere's diameter equating to the equatorial principle of equal division.

Key Concepts

Geometry

Geometry is the branch of mathematics that deals with the sizes, shapes, and properties of figures and spaces. When we talk about the geometry of a sphere, we think about its perfectly round shape in three-dimensional space. A sphere is defined as the set of all points that are a fixed distance, called the radius, from a central point. In this case, the sphere has a radius of \(a\).

Imagine cutting this sphere with two parallel planes that are \(h\) units apart. These planes slice through the sphere, creating a shape known as a spherical cap. A spherical cap is like the top part of a sphere that has been chopped off. The challenge is to find the area of this curved surface between the two planes.

• The sphere is a 3D shape with all points equidistant from the center.
• A spherical cap is a part of the sphere that is cut off by a plane.
• The distance between planes \(h\) is key to determining the cap's height.

Understanding this geometry sets the stage for calculating the area of these fascinating shapes!

Surface Area Formula

The surface area formula for a spherical cap is an important aspect of geometry. To find the area of the cap that is formed when the sphere is sliced by two planes \(h\) units apart, we use a specific formula. The formula for the surface area of a spherical cap depends on both the radius of the sphere and the height of the cap. It is given by:

\[2 \pi a h\]This elegant formula reveals that the area is directly proportional to both the height of the slice \(h\) and the radius \(a\) of the sphere.

• The formula \(2 \pi a h\) gives the surface area of the cap.
• It is derived using principles of calculus and geometry.
• The area is proportional to the cap height \(h\).

By applying this formula, we can easily find the curved surface area of any spherical shape part.

Cylinder and Sphere Relation

The relationship between a cylinder and a sphere is a fascinating exploration into geometry and spatial reasoning. When you circumscribe a cylinder around a sphere, the cylinder's diameter equals the sphere's diameter, meaning the height of the cylinder is twice the radius of the sphere.

Now, consider having a cylinder with a sphere snugly fitted inside. If we imagine slicing both using parallel planes as described, the interesting aspect is that both the spherical cap and the cylindrical portions have the same area. This is due to the intriguing geometrical property that both the sphere and the cylinder share.

• A cylinder can be perfectly sized to encase a sphere.
• The cylindrical height equals the sphere's diameter \(2a\).
• Both have the same surface area when sliced parallel at equal heights.

Understanding these parallels in geometry helps underscore the elegant balance and symmetry in mathematical shapes.

Calculus

Calculus is a powerful mathematical tool used to analyze changes and it plays a crucial role in deriving the surface area of a spherical cap. By breaking down complex problems into infinitely small parts, calculus allows us to compute things that appear difficult at first glance.

To derive the spherical cap's area, we look at the problem using integration. Think of the sphere as being sliced infinitely into rings of small thickness. Each ring is an elementary strip that contributes to the total area. By summing up the areas of all these rings, through integration, we end up with the formula \(2 \pi a h\).

• Calculus uses integration to handle continuous changes.
• It helps find areas by adding up infinitely small parts.
• The formula \(2 \pi a h\) comes from integrating these element strips.

This use of calculus highlights its indispensable role in solving real-life geometry problems and understanding the broader context.