적분법 전체 문제

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

[T] $y=\frac{1}{{x}^{2}}$ from $x=1\,\text{to}\,x=3$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y=\sqrt{4-{x}^{2}}$ from $x=0\,\text{to}\,x=2$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y=\sqrt{4-{x}^{2}}$ from $x=-1\,\text{to}\,x=1$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y=5x$ from $x=1\,\text{to}\,x=5$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

[T] $y=\tan \,x$ from $x=-\frac{\pi }{4}\,\text{to}\,x=\frac{\pi }{4}$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y={x}^{2}$ from $x=0\,\text{to}\,x=2$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y=\frac{1}{2}{x}^{2}+\frac{1}{2}$ from $x=0\,\text{to}\,x=1$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y=x+1$ from $x=0\,\text{to}\,x=3$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

[T] $y=\frac{1}{x}$ from $x=\frac{1}{2}$ to $x=1$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y=\sqrt[3]{x}$ from $x=1\,\text{to}\,x=27$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

[T] $y=3{x}^{4}$ from $x=0$ to $x=1$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

[T] $y=\frac{1}{\sqrt{x}}$ from $x=1$ to $x=3$

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y\text{-axis}.$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

[T] $y=\cos \,x$ from $x=0$ to $x=\frac{\pi }{2}$

The base of a lamp is constructed by revolving a quarter circle $y=\sqrt{2x-{x}^{2}}$ around the $y\text{-axis}$ from $x=1$ to $x=2,$ as seen here. Create an integral for the surface area of this curve and compute it.

A light bulb is a sphere with radius $1/2$ in. with the bottom sliced off to fit exactly onto a cylinder of radius $1/4$ in. and length $1/3$ in., as seen here. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is $1/4$ in. Find the surface area (not including the top or bottom of the cylinder).

[T] A lampshade is constructed by rotating $y=1/x$ around the $x\text{-axis}$ from $y=1$ to $y=2,$ as seen here. Determine how much material you would need to construct this lampshade—that is, the surface area—accurate to four decimal places.

[T] An anchor drags behind a boat according to the function $y=24{e}^{-x/2}-24,$ where $y$ represents the depth beneath the boat and $x$ is the horizontal distance of the anchor from the back of the boat. If the anchor is $23$ ft below the boat, how much rope do you have to pull to reach the anchor? Round your answer to three decimal places.

[T] You are building a bridge that will span $10$ ft. You intend to add decorative rope in the shape of $y=5|\sin ((x\pi )/5)|,$ where $x$ is the distance in feet from one end of the bridge. Find out how much rope you need to buy, measured in a whole number of feet.

For the following exercises, find the exact arc length for the following problems over the given interval.

$y=\ln (\sin \,x)$ from $x=\pi /4$ to $x=(3\pi )/4.$ (Hint: Recall trigonometric identities.)

[T] Draw graphs of $y={x}^{2},$ $y={x}^{6},$ and $y={x}^{10}.$ For $y={x}^{n},$ as $n$ increases, formulate a prediction on the arc length from $(0,0)$ to $(1,1).$ Now, compute the lengths of these three functions and determine whether your prediction is correct.