Any function can be approximated with rectangles.
The smaller the rectangles, the closer the approximation.
Suppose we want to calculate the volume of this object.
Let us first focus on just one slice of the object.
This slice can be thought of as a group of rectangles.
Each of these rectangles has a height of Z and a width of dX.
Therefore, the area of each of these rectangles is Z multiplied by dX.
The total area is the sum of the areas of all these rectangles.
The sum is represented with the following symbol.
The thickness of the slice is dY.
The thickness multiplied by the area of the slice is equal to the slice’s volume.
The area is given by the equation we saw earlier.
Similarly, we can calculate the volume of
each of the different slices that make up the object.
The total volume of the object is the
sum of the volumes of all these individual slices.
The sum is represented with the same symbol as before.
This is one of the ways to calculate the volume of this object.
Suppose that we now instead consider a slice in this other direction.
This time, the area of each rectangle is Z multiplied by dY.
The total area of this slice is the sum of the areas of all these rectangles.
This area multiplied by the thickness is equal to this slice’s volume.
The total volume of the object is the sum
of the volumes of all these individual slices.
Both of the two methods for calculating the volume
end up producing the exact same result.
So far, we have been using what we call rectangular coordinates.
However, there are some objects which are easier to describe with polar coordinates.
In polar coordinates, we still have the same Z axis that we had before.
Instead of using “X” and “Y”, each point is now specified by
how far away it is along the radius from the center…
...and by the angle θ.
As before, suppose that we want to know the volume of this object.
We can start out by focusing on just one slice.
Each of these sections have a height of Z.
And they each have a length of dR.
But, the width of each of these sections is different.
As the radius increases, the arc length also increases.
If the angle changes by dθ, then the arc length changes by R multiplied by dθ.
If dθ and dR are very small, then each of these sections can be approximated by
a rectangle with the following volume.
The total volume of this slice is equal to
the sum of the volumes of each of these sections.
The volume of the entire object is equal to
the sum of the volumes of each of the slices.
This is one of the ways to calculate the volume of this object.
Suppose that we focus on just this one ring.
The volume of each section of the ring has the same formula that was shown before.
The volume of the entire ring is the sum of the volumes of each of the sections.
The volume of the entire object is the sum of the volumes of each of the rings.
Both of the two methods for calculating the volume of this object
end up producing the exact same result.
Let’s calculate the volume of a disk which always has a height exactly equal to one.
Let’s first focus on just one slice of the disk.
If dθ is small, then this slice is approximately a triangle,
and we have therefore produced the equation for the area of this triangle.
To add up all the slices, we must take all the possible values of θ
between zero and 2π.
And hence we have produced the equation for the area of a circle.
Much more information about mathematics
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