Problem 57 - Calculus of a Single Variable

Question

Using a Sphere Find the area of the zone of a sphere formed by revolving the graph of \(y=\sqrt{9-x^{2}}, 0 \leq x \leq 2\)about the \(y\) -axis.

Step-by-Step Solution

Answer

The area of the zone of the sphere is \(6\pi (3 - \sqrt{5})\) units square.

Step 1 Title

Firstly, find the radius of the sphere. This is obtained by setting \(x=0\) in the equation for the circle, which gives \(y=\sqrt{9} = 3\). The radius of the sphere is thus 3.

Step 2 Title

Next, find the limits of the zone of the sphere. The lower limit, \(h1\) is the \(y\)-coordinate when \(x = 0\). Therefore, \(h1 = \sqrt{9-0^{2}} = 3\). The upper limit, \(h2\) is the \(y\)-coordinate when \(x = 2\). Hence, \(h2 = \sqrt{9-2^{2}} = \sqrt{5}\).

Step 3 Title

Substitute \(h1\), \(h2\) and \(r\) (radius = 3) into the formula for the surface area of a frustum of a sphere this is given by: \( A= 2 \pi r (h1 - h2)\). Therefore, the area of the zone becomes \(A = 2 \pi * 3*(3-\sqrt{5})\)

Key Concepts

Solid of Revolution

The concept of a solid of revolution is central to understanding how to calculate the area of spherical zones and many other 3-dimensional shapes. It describes a solid figure obtained by rotating a plane curve around an axis that lies on the same plane. In our exercise, the graph of the semi-circle defined by the equation \(y=\sqrt{9-x^{2}}\) is rotated around the y-axis. This rotation results in a sphere with different zones, determined by the x-coordinates. By understanding this concept, you will be able to visualize how certain volumes and areas in three-dimensional space can be derived from simple two-dimensional curves.

Surface Area of a Sphere

When it comes to the surface area of a sphere, the total area is given by \(4\pi r^{2}\), where \(r\) is the radius of the sphere. However, if we're focusing on just a part of the sphere, as in our problem, we refer to this part as a spherical zone or spherical cap, which is a portion of the sphere cut off by two parallel planes. The formula for the area of a spherical zone is slightly more complex. It involves the radius of the sphere and the heights from the sphere's center to the parallel planes that create the cap. By applying this formula correctly to a specific situation, you can find the exact area of any partial surface of a sphere.

Volume of a Sphere

The volume of a sphere is connected to its surface area and is important in understanding the properties of spherical shapes in three-dimensional space. The volume is determined by the formula \(\frac{4}{3}\pi r^{3}\), where \(r\) again represents the radius. Although we aren't directly looking for volume in this exercise, many problems require you to find the volume of the solid created by the rotation around an axis. Knowing the volume formula can help in understanding the spatial relation between different parts of the sphere and how they relate to the shape as a whole.

Integration in Polar Coordinates

Integration in polar coordinates is a powerful tool to calculate areas and volumes of shapes that have a natural circular symmetry, like spheres and cones. When dealing with curves defined in polar coordinates, you can integrate over angles and radii to find the areas they enclose. This type of integration is crucial when working with solid figures generated by revolving curves around an axis (solids of revolution). In practice, polar coordinates facilitate the calculation of complex shapes that would be more difficult to handle with Cartesian coordinates. When used wisely, it aids in breaking down intimidating geometric problems into approachable components.