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Craville Studies >>
Maths Extension 2 >>
Volumes
Volumes Guide
Volume Types:
Volumes with
Circular Cross Sections
Volumes using
Cylindrical Shells
Volumes
with Parallel Cross-sections of Similar Shape
Volumes with Circular
Cross Sections:
From this revolution we take a typical slice, which
in this case is a spherical slice.
So, therefore the area of the circle can be expressed
as:
Meaning the volume of this slice is:
By summing together and taking the limit, the volume
of the whole solid is:
Since y2 = x
This method can also be applied to rotations around
the y-axis.
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Volumes using Cylindrical
Shells:
This is used when a graph is being rotated about the
y-axis between x = a & x = b.
Take for instance, this example, where we rotate the
area enclosed between the line y = -x + 2 and the x & y-axes.
We firstly take our typical slice. This is a cylinder
with:
Radius = x
Height = y
Width = ∆x
This is
then unfolded to give a rectangle.
Therefore:
Taking the limit and integrating, gives:
Finding definite integral gives:
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Volumes
with Parallel Cross-sections of Similar Shape:
We are given the dimensions of the base and told the
height is 4m. That is, 4m right through the middle of the pyramid
from top to bottom.
For this our typical slice is:
Since the sizes of a, b and c will vary proportional
to how far along the pyramid we go, we break them down into three
single triangles.
This way, each of the values for a, b and c can be
calculated in terms of x.
Since each of these vary proportionally with x:
We now have values for our typical slice.
We then use the area of a triangle.
Meaning the volume of the typical slice is:
By summing and taking the limit:
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