第6章積分の応用

The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

If the force is constant, the amount of work to move an object from $x=a$ to $x=b$ is $F\left(b-a\right).$

The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.

If the half-life of $\text{seaborgium-}266$ is $360$ ms, then $k=\left(\text{ln}(2)\right)\text{/}360.$

The volume that has a base of the ellipse $x^2\text{/}4+y^2\text{/}9=1$ and cross-sections of an equilateral triangle perpendicular to the $y\text{-axis}\text{.}$ Use the method of slicing.

The region bounded by the curve $y=x^2–x$ and the x-axis from $x=1\text{to}x=4,$ rotated around the y-axis using the washer method

$x=y^2$ and $x=3y$ rotated around the y-axis using the washer method

$x=2y^2-y^3,x=0,\text{and}y=0$ rotated around the x-axis using cylindrical shells

[T] $y=\text{ln}\left(x\right)+2\text{and}y=x$

$y=x^2$ and $y=\sqrt{x}$

$y=5+x,$ $y=x^2,$ $x=0,$ and $x=1$

Below $x^2+y^2=1$ and above $y=1-x$

Find the mass of $\rho =e^{\text{−}x}$ on a disk centered at the origin with radius $4.$

Find the center of mass for $\rho ={\text{tan}}^{2}x$ on $x\in \left(-\frac{\pi }{4},\frac{\pi }{4}\right).$

Find the mass and the center of mass of $\rho =1$ on the region bounded by $y=x^5$ and $y=\sqrt{x}.$

The length of $x$ for $y=\text{cosh}\left(x\right)$ from $x=0\text{to}x=2.$

The length of $y$ for $x=3-\sqrt{y}$ from $y=0$ to $y=4$

The shape created by revolving the region between $y=4+x,$ $y=3-x,$ $x=0,$ and $x=2$ rotated around the y-axis.

The loudspeaker created by revolving $y=1\text{/}x$ from $x=1$ to $x=4$ around the x-axis.

For this exercise, consider the Karun-3 dam in Iran. Its shape can be approximated as an inverted isosceles triangle spanning across the river, with height 205 m and width (across the top of the dam) 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m³. Find the total force on the wall of the dam.

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and $45\text{°}\text{F}$ outside and the temperature of the body is $78\text{°}\text{F}.$ You know the cooling constant is $k=0.00824{\text{min}}^{-1}\text{.}$ When did the victim die, assuming that a human’s temperature is $98\text{°}\text{F}$ ?

[T] The best-fit exponential curve to these data is given by $y\approx 48.70656754\times {1.223961577}^x$ Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in $2014$ ?