7.3 Volumes
Calculus
7.3 VOLUMES
Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up
the areas of an infinite number of infinitely thin rectangles, we are going to apply the same concept to finding
volume.
The key … Find the volume of ONE arbitrary "slice", and use an integral to add up the volumes of an
infinite number of infinitely thing "slices".
We will first apply this concept to the volume of a solid with a known cross section, then we will find the volumes
of solids formed by revolving a region about a horizontal or vertical line.
We will discuss three different methods of
finding volumes of solids of revolution, but first …
Day 1:
Volumes of Solids with Known Cross Sections
First Question … What is a cross section?
Imagine a loaf of bread.
Now imagine the shape of a slice through the
loaf of bread.
This shape would be a cross section.
Technically a
cross section
of a three dimensional figure is the
intersection of a plane and that figure.
It would be like cutting an object and then looking at the face of where you
just cut.
The cross sections we will be dealing are almost entirely perpendicular to the
x
– axis.
Here's the basic idea … You will be given a region defined by a number of functions.
We will graph that region on
an
x
and
y
axis.
Then we will lay they region flat and build upon that region a solid which has the same cross
section no matter where you slice it.
To see some animated views of this go to
http://astro.temple.edu/~dhill001/sectionmethod/sectiongallery.html
(We will watch a few of them in class)
Second question … How do we find the volume of this solid that has been created to have a similarly shaped cross
section, even though each cross section may have a different size?
We get to use calculus, of course!
But first, we
need to know how to find the Volume of a prism.
Even though every shape may be different, we can find the
volume of a prism by finding the area of the base times the "height".
The "height" of our prisms will be the
thickness of the slices.
Once you know the volume of one slice, you just use an integral to add the volumes of all
the slices to get the volume of the solid.
Example
:
Find the volume of the following square "slice".
Since most of the "slices" we will be dealing with will
have a thickness of
dx
, we will use that same thickness here.
x
dx
Example
:
Find the volume of the following semicircular "slice".
x
dx
157

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