the calculus crusaders volume
THE CALCULUS CRUSADERSVolumes: The Animal Turd
purple mushrooms by Flickr user yewenyi
THE SITUATIONJamie’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.
Part AZeph 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 g(x) intersect.
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
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.
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.
THE SOLUTIONWe 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:
Part BZeph’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 DIAGRAM
Imagine taking a cylindrical shell and opening it up. We obtain a triangular prism sort of shape. V = 2πr f(x) dxWhere dx, the width, is infinitesimally small so that the shape becomes a rectangular prism. 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 cross-section of the cylinder, we see a hole.
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.
The formula for integrating cylindrical shells:
Part CThe 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 shown.
Recall that the formula for the volume of a triangle can be determined by multiplying the area of the triangular face by the thickness.
In this case, the thickness is infinitesimally small (dx).
The base is the distance between where A(x) is bounded.
THE SOLUTIONTo determine the height of the triangular face, we use trigonometric ratios.
•By totaling the volume of the infinite triangular cross-sections, we obtain the total volume.
!! 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