넓이(여러 가지 함수)

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

$y={x}^{3},x=0,y=0,\,\text{and}\,x=2$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

$y={x}^{2}-x\,\text{and}\,x=0$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

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

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

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

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

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

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

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

Find the mass of $\rho =\frac{1}{{x}^{2}+1}$ on a disk centered at the origin with radius $4.$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

Find the center of mass for $\rho ={\tan }^{2}x$ 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 $y=\sqrt{x}.$

For the following exercises, find the requested arc lengths.

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

For the following exercises, find the requested arc lengths.

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 $y=1/x$ 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 $45^\circ \text{F}$ outside and the temperature of the body is $78^\circ \text{F}.$ You know the cooling constant is $k=0.00824^\circ \text{F/min}.$ When did the victim die, assuming that a human’s temperature is $98^\circ \text{F}$ ?

[T] The best-fit exponential curve to these data is given by $y=40.71+{1.224}^{x}.$ 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 $2014$ ?

Find the volume of the catenoid $y=\cosh (x)$ from $x=-1\,\text{to}\,x=1$ that is created by rotating this curve around the $x\text{-axis},$ as shown here.

Find surface area of the catenoid $y=\cosh (x)$ from $x=-1$ to $x=1$ that is created by rotating this curve around the $x\,\text{-axis.}$

T/F: The area between curves is always positive.

T/F: Calculus can be used to find the area of basic geometric shapes.