THE CALCULUS CRUSADERS
Volumes: The Animal Turd
purple mushrooms by Flickr user yewenyi
Jamie’s duck foolishly ate the wild mushroom!
Thankfully the duck defecated on the sand and
got rid of the ache in its stomach.
Zeph oddly notices that the turd covers a region
of the sand equivalent to the shaded
region, R, shown in the graph. He also imagines
a Cartesian plane behind the turd.
A(x) is the region bounded by the function f(x) = 1/x and
g(x) = sin(x), measured in cm2.
a) Zeph wants to collect some data about the turd. Determine the
area of A(x).
b) Zeph’s koala likes to get dirty. He smears the turd around the y-
axis. Determine the volume of the solid when A(x) is revolved
about the y-axis.
c) The region A(x) is the base of a solid, where each cross-section
perpendicular to the x-axis is an equilateral triangle. Find the
volume of this solid.
Zeph wants to collect some data about the turd. Determine
the area of A(x).
Determining an area underneath a graph is the
definition of integration, but we must first know
the upper and lower limits—the interval at which
we are integrating.
Looking at the graph, we see that we have to
integrate between two points at which f(x) and
Since is a transcendental function, a function
that contains an exponential function and a
trigonometric function, we cannot apply the
algebra we know to solve for the roots of v’(t), so
we have to use our calculator to solve
x = 1.1141571, 2.7726047
Points of intersection at x =
To make our work look less cluttered, we can
assign unappealing numbers to letters;
▫ Let S = 1.1141571
▫ Let T = 2.7726047.
Of course, functions f(x) and g(x) intersect at other
places too, such as the area bounded by f(x) and g(x) in
the second quadrant near the y-axis as shown in the
graph given, but we are only interested in the x-
coordinates where R is bounded.
We integrate the top function, sin x, from S to T.
We integrate the bottom function, 1/x, from S to T.
Take the difference, “TOP” function minus “BOTTOM”,
to obtain A(x). This is represented by:
Zeph’s koala likes to get dirty. He smears the turd around
the y-axis. Determine the volume of the solid when A(x) is
revolved about the y-axis.
Revolving around the y-axis generates a cylinder.
We can imagine there are infinite cylindrical shells.
Getting the total of the shells would give us the total
volume by the definition of integration.
THE TISSUE PAPER ROLL
Cylinder Imagine taking a cylindrical
shell and opening it up.
We obtain a triangular prism
sort of shape.
V = 2πr f(x) dx
Where dx, the width, is
Prism infinitesimally small so that
the shape becomes a
This is similar to unraveling
tissue paper from it’s roll.
Again, a cylindrical shell would have a volume of
2πr f(x)dx, where 2πr is the length, f(x) is the
height, and dx is the width/thickness of prism.
**(Recall that the formula for the volume of a cylinder is V(x) = 2πr2h.
Note the similarities.)
So, its radius becomes x (as well as the distance
away from the y-axis if we are revolving the
area around a line other than the y-axis).
Looking at a
the cylinder, we
see a hole.
This means that the
cylinder is hollow at
its centre, and the
height of the
shell, f(x), is the
minus the lower
The formula for integrating cylindrical shells:
The region A(x) is the base of a solid, where each cross-
section perpendicular to the x-axis is an equilateral
triangle. Find the volume of this solid.
An equilateral triangle
is defined as a three-
sided shape with three
congruent sides and
three congruent angles.
A cross-section is
Recall that the
formula for the
volume of a triangle
can be determined
by multiplying the
area of the
triangular face by
In this case, the
The base is the
where A(x) is
To determine the
height of the
triangular face, we
• By totaling the volume of the infinite triangular
cross-sections, we obtain the total volume.
Jamie’s duck has
WE’VE DONE IT!!
taken an.. Erk and
we’re all happy
and can continue
on our journey!!
The Happy little Duck by Flickr