Project 7: Applications of Integration

Instructions

Use Maxima to work this project. Each group should email an annotated Maxima file to the instructor.

Arclength

1. Corrugated Steel Panels

Menards sells corrugated galvanized steel panels. They are made out of 30 gauge galvanized steel, are 26 inches wide, 8 feet long, and are ribbed to be 1/2 inch thick. The ribs are 2.5 inches on-center. Although not exact, a cross-section slice can be modeled by \( y = \frac 1 2 \sin ( 0.8 \pi x ),\ 0\le x \le 26\).

How many lineal inches of steel are required to produce the 26 inch wide panel? That is, how wide was the steel before it was pressed? In calculus terms, that's the arclength of the curve.
What is the weight in pounds of a panel? Hint: Search for the weight density of 30 gauge galvanized steel.

Theorems of Pappus

There are two theorems attributed to Pappus of Alexandria and Paul Guldin (they did not co-author them). They allow you to find surface areas and volumes of three-dimensional objects found by rotating two dimensional objects.

The area of a surface generated by rotating a curve about an axis, which does not pass through the curve, is the length of the curve times the distance traveled by the centroid of the curve.
The volume of a solid generated by rotating a region about an axis, which does not pass through the region, is the area of the region times distance traveled by the centroid of the region.

2. Theorem of Pappus for a Curve

Consider the curve \( f(x) = \displaystyle \frac 1 2 (x-2)^2 \) on \( 0 \le x \le 6 \).

Graph the curve.
Find the length of the curve.
Find the centroid of the curve.
Use the integral for surface area of revolution to find the area of the surface generated by rotating the curve about the x-axis. Then verify the result using the Theorem of Pappus.
Use the easiest method to find the area of surface generated by rotating the curve about the y-axis.
Use the easiest method to find the area of surface generated by rotating the curve about the line \( 2x - y = 8 \).

3. Theorem of Pappus for a Region

Consider the region bounded by the curves \( f(x) = \displaystyle \frac 1 2 (x-2)^2 \) and \( g(x) = x + 2 \).

Graph the region.
Find the area of the region.
Find the centroid of the region.
Use the disk, washer, or cylindrical shells method (whichever is appropriate), to find the volume of the solid generated by rotating the region about the x-axis. Then verify the result using the Theorem of Pappus.
Use the disk, washer, or cylindrical shells method (whichever is appropriate), to find the volume of the solid generated by rotating the region about the y-axis. Then verify the result using the Theorem of Pappus.
Use the disk, washer, or cylindrical shells method (whichever is appropriate), to find the volume of the solid generated by rotating the region about the line \( x = -1 \). Then verify the result using the Theorem of Pappus.
Use the disk, washer, or cylindrical shells method (whichever is appropriate), to find the volume of the solid generated by rotating the region about the line \( y = 9 \). Then verify the result using the Theorem of Pappus.
Use the easiest method to find the volume of the solid generated by rotating the region about the line \( 2x - y = 8 \).