적분의 활용 전체 문제

For the following exercises, compute the center of mass $\overset{-}{x}.$

$\rho ={e}^{x}$ for $x\in (0,2)$

For the following exercises, compute the center of mass $\overset{-}{x}.$

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

For the following exercises, compute the center of mass $\overset{-}{x}.$

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

For the following exercises, compute the center of mass $\overset{-}{x}.$

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

For the following exercises, compute the center of mass $\overset{-}{x}.$

$\rho =\ln \,x$ for $x\in (1,e)$

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

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

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

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

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

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

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 $y=\cos (2x),$ $x=-\frac{\pi }{4},$ and $x=\frac{\pi }{4}$

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 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?