1. THE CALCULUS CRUSADERS
Volumes: The Animal Turd
Question
purple mushrooms by Flickr user yewenyi

2. THE SITUATION
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.

3. A(x) is the region bounded by the function f(x) = 1/x and
g(x) = sin(x), measured in cm2.

4. THE QUESTION
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.

5. Zeph wants to collect some data about the turd. Determine
the area of A(x).

6. THE SOLUTION
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.

7. THE SOLUTION
Looking at the graph, we see that we have to
integrate between two points at which f(x) and
g(x) intersect.

8. THE SOLUTION
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
numerically.
x = 1.1141571, 2.7726047

9. THE SOLUTION
Points of intersection at x =
1.1141571, 2.7726047.
To make our work look less cluttered, we can
assign unappealing numbers to letters;
▫ Let S = 1.1141571
▫ Let T = 2.7726047.

10. THE SOLUTION
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.

11. THE SOLUTION
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:

12. 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.

13. THE SOLUTION
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.

14. THE TISSUE PAPER ROLL
DIAGRAM
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 rectangular prism.
This is similar to
unraveling tissue paper
from it’s roll.

15. THE SOLUTION
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.)

16. THE SOLUTION
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).

17. Looking at a
cross-section of
the cylinder, we
see a hole.

18. This means that the
cylinder is hollow at
its centre, and the
height of the
cylindrical
shell, f(x), is the
upper function
minus the lower
function.

19. THE SOLUTION
The formula for integrating cylindrical shells:

20. 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.

21. THE SOLUTION
An equilateral
triangle is defined
as a three-sided
shape with three
congruent sides and
three congruent
angles. A cross-
section is shown.

22. THE SOLUTION
Recall that the
formula for the
volume of a triangle
can be determined
by multiplying the
area of the
triangular face by
the thickness.

23. THE SOLUTION
In this case, the
thickness is
infinitesimally
small (dx).

24. THE SOLUTION
The base is the
distance between
where A(x) is
bounded.

25. THE SOLUTION
To determine the
height of the
triangular face, we
use trigonometric
ratios.

26. THE SOLUTION
• By totaling the volume of the infinite triangular
cross-sections, we obtain the total volume.

27. WE’VE DONE IT!!
Jamie’s duck has taken an.. Erk
and we’re all happy and can
continue on our journey!!
The Happy little Duck by Flickr
user law_keven