Problem 9 - The Calculus with Analytic Geometry

Question

\text { Find the volume of the sphere generated by revolving the circle whose equation is } x^{2}+y^{2}=r^{2} \text { about a diameter. }

Step-by-Step Solution

Answer

\(\frac{4}{3} \pi r^{3}\)

Step 1 - Understand the Problem

Given the equation of the circle is \(x^{2}+y^{2}=r^{2}\). This circle is being revolved around its diameter to form a sphere.

Step 2 - Identify the Shape

Revolving a circle around its diameter creates a sphere. The radius of the sphere is the same as the radius of the circle, denoted by \(r\).

Step 3 - Formula for Volume of a Sphere

The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^{3}\).

Step 4 - Substitute the Radius

Substitute the radius \(r\) from the given circle's equation into the volume formula: \(V = \frac{4}{3} \pi r^{3}\).

Step 5 - Final Volume Calculation

Since the radius \(r\) is already given, the final volume \(V\) of the sphere is \(\frac{4}{3} \pi r^{3}\).

Key Concepts

equation of a circle

The equation of a circle in the standard form is usually written as: \(x^{2}+y^{2}=r^{2}\).
Here, \(r\) stands for the radius of the circle. This equation represents all the points \((x, y)\) that are exactly \(r\) units away from the center of the circle, which is at the origin \((0, 0)\).
Understanding this equation is crucial when we need to compute areas, lengths, or when we transform the circle into another shape by processes like revolving it.

revolving a shape

When we revolve a shape around an axis, we generate a 3-dimensional object. In this problem, we revolve a circle around its diameter.
This specific transformation creates a sphere. All points on the circumference of the circle trace out circles themselves, forming the surface of the sphere.
In our case, if the circle has a radius \(r\), the sphere generated will have the same radius \(r\). Studying revolving helps us understand how 2D shapes can turn into 3D objects.

geometry formulas

Geometry provides multiple formulas to calculate different properties of shapes. For a sphere, one of the most important formulas is the volume formula:

• The volume formula for a sphere is \(V = \frac{4}{3} \pi r^{3}\).

Here, \(\pi\) is a mathematical constant approximately equal to 3.14159, and \(r\) is the radius of the sphere.
In this problem, we simply substitute the value of \(r\) given by the radius of the circle (from the equation \(x^{2}+y^{2}=r^{2}\)) into the volume formula to find the final answer.