곡선의 길이

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.

Compare the lengths of the parabola $x={y}^{2}$ and the line $x=by$ from $(0,0)\,\text{to}\,({b}^{2},b)$ as $b$ increases. What do you notice?

Solve for the length of $x={y}^{2}$ from $(0,0)\,\text{to}\,(1,1).$ Show that $x=(1/2){y}^{2}$ from $(0,0)$ to $(2,\,2)$ is twice as long. Graph both functions and explain why this is so.

[T] Which is longer between $(1,\,1)$ and $(2,1/2):$ the hyperbola $y=1/x$ or the graph of $x+2y=3?$

Explain why the surface area is infinite when $y=1/x$ is rotated around the $x\text{-axis}$ for $1\le x<\infty ,$ but the volume is finite.

T/F: The integral formula for computing Arc Length was found by first approximating arc length with straight line segments.

T/F: The integral formula for computing Arc Length includes a square–root, meaning the integration is probably easy.

In Exercises 3–12, find the arc length of the function on the given interval.

$\displaystyle f(x) = x$ on $[0, 1]$.

In Exercises 3–12, find the arc length of the function on the given interval.

$\displaystyle f(x) = \sqrt{8}x$ on $[-1, 1]$.

In Exercises 3–12, find the arc length of the function on the given interval.

$\displaystyle f(x) = \frac13x^{3/2}-x^{1/2}$ on $[0,1]$.

In Exercises 3–12, find the arc length of the function on the given interval.

$\displaystyle f(x) = \frac1{12}x^{3}+\frac1x$ on $[1,4]$.

In Exercises 3–12, find the arc length of the function on the given interval.

$\displaystyle f(x) = 2x^{3/2}-\frac16\sqrt{x}$ on $[0,9]$.

In Exercises 3–12, find the arc length of the function on the given interval.

$\displaystyle f(x) = \cosh x$ on $[-\ln 2, \ln 2]$.