` Volumes Of Simple Solids

Volumes of Simple Solids

We consider here different methods for computing the volumes of two 3-dimensional solids: a sphere (a ball) and a cylinder. We have simple algebraic formulas for their volumes, but we are going to look at another method that is more complex but also much more general. We'll see that it also allows us to compute volumes of solids for which we do not have algebraic formulas.

A ball
The volume V of a ball having radius R is

V = 4/3πR³.

But now we consider a different way of computing it. Let's imagine that we put the X-axis along one of its diameters, and mark the distance from the center of the ball. Let's cut the ball into very thin slices of width dX (like slices of a tomato).

The area of a cut at a distance X from the center is π(R² - X²). Do you know why? (Use the Pythagorean theorem.) So the volume of a slice next to this cut that has thickness dX is approximately

π(R² - X²)dX

This is so because the shape of the slice is very close to a cylinder with base π(R² - X²) and height dX. Thus, if we add now the volumes of all the slices, we get an approximation of the volume of the ball. Thinner slices (dX close to 0) give better approximations. The mathematical expression that summarizes this method is written as follows

_-R∫^Rπ(R² - X²)dX

Its standard reading is: "The integral, for X changing from -R to R, of π(R² - X²) dee ex."
It means, "Split the segment from -R to R into small pieces dX, compute the products π(R² - X²)dX, and the sum of these products is your answer." The symbol ∫ is just a thin S. It is called the integral because, "We put together many small parts to create a whole. We integrate* the parts."
We have now a new mathematical formula for the volume V of a sphere,

V = _-R∫^Rπ(R² - X²)dX

On the TI-83/84 Plus, you compute it as follows,

fnInt(π(R²-X²),X,-R,R) ENTER

It is read, "the integral (function) of π(R² - X²), relative to X (or dee ex), where X changes from -R to R."

Use both formulas,

V = 4/3πR³ and

fnInt(π(R²-X²),X,-R,R)

to compute the volume of a sphere for a few values of R.
Find the volume of a golf ball using both formulas.

Task 2
A sphere with a radius R = 7 inches was cut into 3 slices as shown below,

Compute their volumes. Try it on your own before looking below!
Solution.
From the homescreen, enter

	7→R				ENTER
	fnInt(π(R²-X²),X,-R,-1/3R)				ENTER
	fnInt(π(R²-X²),X,-1/3R,1/2R)				ENTER
	fnInt(π(R²-X²),X,1/2R,R)				ENTER

Remarks.

	The second task can also be solved by an algebraic formula. And the Fundamental Theorem of Calculus (which we will discuss later) can be used to find it (using antiderivatives). But it is not a topic of this lesson.
	The calculator does not compute the integral directly by the method described above. It uses a very ingenious "short cut" that was invented by Gauss and later improved even more.

Cylinder
The volume V of a cylinder with the radius of its base equal to R, and height equal to H, is

	V = πR²H
The other formula is,
	V = ₀∫^HπR²dX

Here, the slices are circles.
(Can you derive this formula also by using horizontal slices?)

Task.
From a cylinder with radius R = 3 in., and height H = 5 in. a vertical slice of thickness 1 in. was cut off, creating a rectangular opening. Design and construct from poster board such an "open" cylinder, and compute its volume using vertical slices.

We have to write the general formula for the width of the rectangle:

Can you compute the length of the remaining part of the circle that is the cylinder's base?
What percentage of the volume of the cylinder has been cut off?

See the unit Slicing a cylinder revisited if you need some help!

Webpage Maintained by Owen Ramsey
Calculus Index

Volumes of Simple Solids

We have to write the general formula for the width of the rectangle:

Can you compute the length of the remaining part of the circle that is the cylinder's base? What percentage of the volume of the cylinder has been cut off? See the unit Slicing a cylinder revisited if you need some help!

Can you compute the length of the remaining part of the circle that is the cylinder's base?
What percentage of the volume of the cylinder has been cut off?

See the unit Slicing a cylinder revisited if you need some help!