6.6 Moments and centers of mass  (Page 8/14)

$m_1=2$ at $x_1=1$ and $m_2=4$ at $x_2=2$

$m_1=1$ at $x_1=\mathrm{-1}$ and $m_2=3$ at $x_2=2$

$m=3$ at $x=0,1,2,6$

Unit masses at $(x,y)=(1,0),(0,1),(1,1)$

$m_1=1$ at $(1,0)$ and $m_2=4$ at $(0,1)$

$m_1=1$ at $(1,0)$ and $m_2=3$ at $(2,2)$

$\rho =1$ for $x\in (\mathrm{-1},3)$

$\rho =x^2$ for $x\in (0,L)$

$\rho =1$ for $x\in (0,1)$ and $\rho =2$ for $x\in (1,2)$

$\rho =\text{sin}x$ for $x\in (0,\pi )$

$\rho =\text{cos}x$ for $x\in \left(0,\frac{\pi }{2}\right)$

$\rho =e^x$ for $x\in \left(0,2\right)$

$\rho =x^3+x{e}^{\text{−}x}$ for $x\in (0,1)$

$\rho =x\text{sin}x$ for $x\in (0,\pi )$

$\rho =\sqrt{x}$ for $x\in \left(1,4\right)$

$\rho =\text{ln}x$ for $x\in \left(1,e\right)$

$\rho =7$ in the square $0\le x\le 1,$ $0\le y\le 1$

$\rho =3$ in the triangle with vertices $(0,0),$ $(a,0),$ and $(0,b)$

$\rho =2$ for the region bounded by $y=\text{cos}(x),$ $y=\text{−}\text{cos}(x),$ $x=-\frac{\pi }{2},$ and $x=\frac{\pi }{2}$

[T] The region bounded by $y=\text{cos}(2x),$ $x=-\frac{\pi }{4},$ and $x=\frac{\pi }{4}$

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

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

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

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

[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

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

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?

[T] Quarter-circle: $y=\sqrt{1-x^2},$ $y=0,$ and $x=0$

[T] Triangle: $y=x,$ $y=2-x,$ and $y=0$

[T] Lens: $y=x^2$ and $y=x$

[T] Ring: $y^2+x^2=1$ and $y^2+x^2=4$

[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 $y=b\text{sin}(ax),$ $x=0,$ 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.

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