6.6 Calc

1.

\fFind the centroid of the region bounded by the graphs of the functions 1 7T y = 2 sin(x), y = 09 c, and c = , and touching the origin. The centroid is at (@, 7 ) where T. UAn artist has decided to finish their piece of artwork by balancing it on a fulcrum and putting it on display. The artwork has constant density and must be balanced at its centroid. The shape of the artwork was created on a computer program then casted and fabricated. The following equation was put into the computer to generate the shape: y = 1 sin(x ) + 6 bounded by x = 0, x = 2, and y = 0 Draw the Lamina in an x-y plane and put a dot where the centroid should be. Show all work and formulas you are using. The centroid is at (, y ), where T = y =Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a = 3 is positioned with the left end of the circle at c = 6 = 4, and is rotated around the y-axis. a Volume = Write as an exact expression. Use pi for .Use the Theorem of Pappus to find the exact volume of a general cone created by rotating a triangle with vertices (0, 0), (2, 0), and (0, 4) around the y-axis. Volume =