넓이(여러 가지 함수)

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $x={e}^{y}\,\text{and}\,y=x-2$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y={x}^{2}\,\text{and}\,y=\sqrt{1-{x}^{2}}$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y=3{x}^{2}+8x+9\,\text{and}\,3y=x+24$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] ${x}^{2}={y}^{3}\,\text{and}\,x=3y$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y={\sin }^{3}x+2,y=\tan \,x,x=-1.5,\,\text{and}\,x=1.5$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y=\sqrt{1-{x}^{2}}\,\text{and}\,{y}^{2}={x}^{2}$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y=\sqrt{1-{x}^{2}}\,\text{and}\,y={x}^{2}+2x+1$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $x=4-{y}^{2}\,\text{and}\,x=1+3y+{y}^{2}$

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y=\cos \,x,y={e}^{x},x=-\pi ,\,\text{and}\,x=0$

An amusement park has a marginal cost function $C(x)=1000{e}^{-x}+5,$ where $x$ represents the number of tickets sold, and a marginal revenue function given by $R(x)=60-0.1x.$ Find the total profit generated when selling $550$ tickets. Use a calculator to determine intersection points, if necessary, to two decimal places.

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H(t)=1-\cos ((\pi t)/2)$ whereas the speed of the tortoise is $T(t) = 0.1032t$ where $t$ is time measured in hours and the speed is measured in miles per hour. Find the area between the curves from time $t=0$ to the first time after one hour when the tortoise and hare are traveling at the same speed. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H(t)=(1/2)-(1/2)\cos (2\pi t)$ whereas the speed of the tortoise is $T(t)=\sqrt{t},$ where $t$ is time measured in hours and speed is measured in kilometers per hour. If the race is over in $1$ hour, who won the race and by how much? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

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.

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.

For the following exercises, use the requested method to determine the volume of the solid.

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

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 $x=1$