적분의 활용 전체 문제

[T] The force of gravity on a mass $m$ is $F=-((GMm)/{x}^{2})$ newtons. For a rocket of mass $m=1000\,\text{kg},$ compute the work to lift the rocket from $x=6400$ to $x=6500$ km. State your answers with three significant figures. (Note: $G=6.67\,\times \,{10}^{-11}\,{\text{N m}}^{2}/{\text{kg}}^{2}$ and $M=6\,\times \,{10}^{24}\,\text{kg}.)$

[T] For the rocket in the preceding exercise, find the work to lift the rocket from $x=6400$ to $x=\infty .$

[T] A rectangular dam is $40$ ft high and $60$ ft wide. Assume the weight density of water is 62.5 lbs/ft³. Compute the total force $F$ on the dam when

a. the surface of the water is at the top of the dam and b. the surface of the water is halfway down the dam.

[T] Find the work required to pump all the water out of a cylinder that has a circular base of radius $5$ ft and height $200$ ft. Use the fact that the density of water is $62$ lb/ft³.

[T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.

[T] How much work is required to pump out a swimming pool if the area of the base is $800$ ft², the water is $4$ ft deep, and the top is $1$ ft above the water level? Assume that the density of water is $62$ lb/ft³.

A cylinder of depth $H$ and cross-sectional area $A$ stands full of water at density $\rho .$ Compute the work to pump all the water to the top.

For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.

A cone-shaped tank has a cross-sectional area that increases with its depth: $A=(\pi {r}^{2}{h}^{2})/{H}^{3}.$ Show that the work to empty it is half the work for a cylinder with the same height and base.

For the following exercises, calculate the center of mass for the collection of masses given.

${m}_{1}=2$ at ${x}_{1}=1$ and ${m}_{2}=4$ at ${x}_{2}=2$

For the following exercises, calculate the center of mass for the collection of masses given.

${m}_{1}=1$ at ${x}_{1}=-1$ and ${m}_{2}=3$ at ${x}_{2}=2$

For the following exercises, calculate the center of mass for the collection of masses given.

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

For the following exercises, calculate the center of mass for the collection of masses given.

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

For the following exercises, calculate the center of mass for the collection of masses given.

${m}_{1}=1$ at $(1,0)$ and ${m}_{2}=4$ at $(0,1)$

For the following exercises, calculate the center of mass for the collection of masses given.

${m}_{1}=1$ at $(1,0)$ and ${m}_{2}=3$ at $(2,2)$

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

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

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

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

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

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

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

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

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

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