91Ó°ÊÓ

For the following exercises, use the theorem of Pappus to determine the volume of the shape. Rotating \(y=m x\) around the \(y\) -axis between \(x=0\) and \(x=1\)

For the following exercises, use the theorem of Pappus to determine the volume of the shape. A general cone created by rotating a triangle with vertices \((0,0), \quad(a, 0),\) and \((0, b)\) around the \(y\) -axis. Does your answer agree with the volume of a cone?

For the following exercises, use the theorem of Pappus to determine the volume of the shape. A general cylinder created by rotating a rectangle with vertices \((0,0), \quad(a, 0),(0, b),\) and \((a, b)\) around the \(y\) -axis. Does your answer agree with the volume of a cylinder?

For the following exercises, use the theorem of Pappus to determine the volume of the shape. A sphere created by rotating a semicircle with radius \(a\) around the \(y\) -axis. Does your answer agree with the volume of a sphere?

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x}, \bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. [T] Quarter-circle: \(y=\sqrt{1-x^{2}}, \quad y=0,\) and \(x=0\)

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x}, \bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. [T] Triangle: \(y=x, \quad y=2-x,\) and \(y=0\)

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x}, \bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. [T] Lens: \(y=x^{2}\) and \(y=x\)

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x}, \bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. [T] Ring: \(y^{2}+x^{2}=1\) and \(y^{2}+x^{2}=4\)

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x}, \bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. [T] Half-ring: \(y^{2}+x^{2}=1, \quad y^{2}+x^{2}=4,\) and \(y=0\)

Find the center of mass \((\bar{x}, \bar{y})\) for a thin wire along the semicircle \(y=\sqrt{1-x^{2}}\) with unit mass. (Hint: Use the theorem of Pappus.)