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Chapter 6: Problem 12

Finding Surface Area Find the surface area of a sphere of radius \(r\) centered at the origin.

Short Answer

Expert verified
The surface area of the sphere is \(4\pi r^2\).

Step by step solution

01

Understand the Formula

To find the surface area of a sphere, we use the formula: \[ A = 4\pi r^2 \]where \(A\) is the surface area and \(r\) is the radius of the sphere.
02

Substitute the Value of the Radius

Insert the given radius \(r\) into the surface area formula. In this case, the radius is given as \(r\). Thus the expression becomes:\[ A = 4\pi r^2 \]
03

Calculate the Surface Area

Since the radius is denoted by \(r\), and this is a symbolic representation, the expression \(4\pi r^2\) is the final representation of the surface area of the sphere. Use a calculator if you have specific numerical values to plug in for \(r\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometry and the Sphere
The geometry of a sphere is fascinating. A sphere is a perfectly symmetrical 3D shape, where every point on its surface is equidistant from its center.
This distance is known as the radius. A sphere surface doesn't have any edges or vertices like those you might find in a cube or pyramid, meaning it's a smooth shape, just like a ball.
A sphere can be best understood by thinking about everyday objects, like a basketball or a globe.
  • Center: The midpoint inside the sphere.
  • Radius: Straight line from the center to any point on the surface.
  • Diameter: The longest distance from one side of the sphere to the other, passing through the center. It's twice the radius.
Understanding these elements is essential in geometry to work with spheres.
When you think about sphere geometry, visualize these aspects to grasp the calculations better.
Mathematics and the Surface Area
In mathematics, calculating the surface area of a sphere involves using the formula: \[A = 4\pi r^2\]This expression is the result of complex mathematical derivations but can be broken down in simple terms:
  • \(\pi\) (pi) is a mathematical constant roughly equal to 3.14159. It's essential in dealing with circles and spheres.
  • The \(r\) is the radius of the sphere. The formula requires this as it's the crucial dimension that affects the surface area.
  • The square of the radius \(r^2\) accounts for the sphere's 3-dimensional nature. When squared, it describes how changes in the radius affect the overall area.
Mathematics makes it possible to predict and comprehend physical properties like surface area with such formulas.
Using a formula like this can simplify understanding complex shapes and quantifying their dimensions.
Calculus and Derivation
Calculus plays a significant role in deriving the formula for the surface area of a sphere. It involves integration techniques that stem from our need to break down complex surfaces into more manageable parts.
The logic goes:
  • Imagine slicing the sphere into infinitesimally small rings. Calculus principles help sum these rings to calculate the whole surface area.
  • By using calculus, mathematicians pinpoint the infinitesimal changes in the sphere's geometry and integrate these changes to form a holistic formula.
  • This method accurately captures the sphere's continuous surface by considering its smooth curvature.
Using calculus, mathematicians derived the formula \(A = 4\pi r^2\) to model the sphere's surface precisely.
Though calculus is advanced math, it offers powerful tools for understanding and computing areas and volumes in the natural world.

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Most popular questions from this chapter

For the following exercises, sketch the graph of each conic. $$ 25 x^{2}-4 y^{2}=100 $$

Without using technology, sketch the polar curve \(\theta=\frac{2 \pi}{3}\).

For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. $$ \text { Directrix: } x=-4 ; e=5 $$

For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), so the polar equation \(r=f(\theta)\) is now written in parametric form.\(r=2 \sin (3 \theta) ;\) tips of the leaves

For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{7}{5-5 \cos \theta} $$
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