밀도·질량과 질량중심

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region between $y=2{x}^{2},$ $y=0,$ $x=0,$ and $x=1$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region between $y=\frac{5}{4}{x}^{2}$ and $y=5$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] Region between $y=\sqrt{x},$ $y=\ln (x),$ $x=1,$ and $x=4$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by $y=0,$ $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by $y=0,$ $x=0,$ and $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$

[T] The region bounded by $y={x}^{2}$ and $y={x}^{4}$ in the first quadrant

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Rotating $y=mx$ around the $x$-axis between $x=0$ and $x=1$

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Rotating $y=mx$ around the $y$-axis between $x=0$ and $x=1$

A general cone created by rotating a triangle with vertices $(0,0),$ $(a,0),$ and $(0,b)$ around the $y$-axis. Does your answer agree with the volume of a cone?

A general cylinder created by rotating a rectangle with vertices $(0,0),$ $(a,0),(0,b),$ and $(a,b)$ around the $y$-axis. Does your answer agree with the volume of a cylinder?

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 $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Quarter-circle: $y=\sqrt{1-{x}^{2}},$ $y=0,$ 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 $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Triangle: $y=x,$ $y=2-x,$ 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 $\left(\overset{-}{x},\overset{-}{y}\right)$ 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 $\left(\overset{-}{x},\overset{-}{y}\right)$ 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 $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Half-ring: ${y}^{2}+{x}^{2}=1,$ ${y}^{2}+{x}^{2}=4,$ and $y=0$

Find the generalized center of mass in the sliver between $y={x}^{a}$ and $y={x}^{b}$ with $a>b.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Find the generalized center of mass between $y={a}^{2}-{x}^{2},$ $x=0,$ and $y=0.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Find the generalized center of mass between $y=b\,\sin (ax),$ $x=0,$ and $x=\frac{\pi }{a}.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius $a$ is positioned with the left end of the circle at $x=b,$ $b>0,$ and is rotated around the y-axis.