?The figure shows a telephone wire hanging between two poles at x=-25 and x=25. The wire

Use the arc length formula (3) to find the length of the curve \(y=3-2 x,-1 \leqslant x \leqslant 3\). Check your answer by noting that the curve is a line segment and calculating its length by the distance formula. Equation Transcription: ? ? Text Transcription: y = 3 ? 2x, ?1 less than or equal slant x less than or equal slant 3

Use the arc length formula to find the length of the curve \(y=\sqrt{4-x^2}, 0 \leqslant x \leqslant 2\). Check your answer by noting that the curve is part of a circle.

Set up, but do not evaluate, an integral for the length of the curve. \(y=x^{3}, \quad 0 \leqslant x \leqslant 2\) ________________ Equation Transcription: ? ? Text Transcription: y = x^3, 0 less than or equal slant x less than or equal slant 2

Set up, but do not evaluate, an integral for the length of the curve. \(y=e^{x}, \quad 1 \leqslant x \leqslant 3\) Equation Transcription: ? ? Text Transcription: y = e^x, 1 less than or equal slant x less than or equal slant 3

Set up, but do not evaluate, an integral for the length of the curve. \(y=x-\ln x, \quad 1 \leqslant x \leqslant 4\) Equation Transcription: ? ? Text Transcription: y = x ? ln x, 1 less than or equal slant x less than or equal slant 4

Set up, but do not evaluate, an integral for the length of the curve. \(x=y^2+y, \quad 0 \leqslant y \leqslant 3\)

Set up, but do not evaluate, an integral for the length of the curve. x=sin y, \(0 \leqslant y \leqslant \pi / 2\)

Set up, but do not evaluate, an integral for the length of the curve. \(y^{2}=\ln x, \quad-1 \leqslant y \leqslant 1\) Equation Transcription: ? ? Text Transcription: y^2=ln x,-1 less than or equal slant y less than or equal slant 1

Find the exact length of the curve. \(y=\frac{2}{3} x^{3 / 2}, \quad 0 \leqslant x \leqslant 2\) Equation Transcription: ? ? Text Transcription: y=2/3 x^3/2, 0 less than or equal slant x less than or equal slant 2

Find the exact length of the curve. \(y=(x+4)^{3 / 2}, \quad 0 \leqslant x \leqslant 4\) Equation Transcription: ? ? 4 Text Transcription: y=(x+4)^3/2, 0 less than or equal slant x less than x less than or equal slant x less than 4

Find the exact length of the curve. \(y=\frac{2}{3}\left(1+x^{2}\right)^{3 / 2}, \quad 0 \leqslant x \leqslant 1\) Equation Transcription: ?? Text Transcription: y=2/3(1+x^2)^3/2, 0 less than or equal slant x less than or equal slant 1

Find the exact length of the curve. \(36 y^{2}=\left(x^{2}-4\right)^{3}\), \(2 \leqslant x \leqslant 3\), \(y \geqslant 0\)

Find the exact length of the curve. \(y=\frac{x^{3}}{3}+\frac{1}{4 x}\), \(1 \leqslant x \leqslant 2\)

Find the exact length of the curve. \(x=\frac{y^{4}}{8}+\frac{1}{4 y^{2}}, \quad 1 \leqslant y \leqslant 2\) Equation Transcription: ?? Text Transcription: x=y^4/8+1/4y^2, 1 less than or equal slant y less than or equal slant 2

Find the exact length of the curve. \(y=\frac{1}{2} \ln (\sin 2 x), \quad \pi / 8 \leqslant x \leqslant \pi / 6\) Equation Transcription: ? ? Text Transcription: y=1/2 ln(sin 2x), pi/8 less than or equal slant x less than or equal slant pi/6

Find the exact length of the curve. \(y=\ln (\cos x)\), \(0 \leqslant x \leqslant \pi / 3\)

Find the exact length of the curve. \(y=\ln (\sec x), \quad 0 \leqslant x \leqslant \pi / 4\) Equation Transcription: ?? Text Transcription: y=ln(sec x), 0 less than or equal slant x less than or equal slant pi/4

Find the exact length of the curve. \(x=e^{y}+\frac{1}{4} e^{-y}, \quad 0 \leqslant y \leqslant 1\) Equation Transcription: ? ? Text Transcription: x=e^y+1/4e^-y, 0 less than or equal slant y less than or equal slant 1

Find the exact length of the curve. \(x=\frac{1}{3} \sqrt{y}(y-3), \quad 1 \leqslant y \leqslant 9\) Equation Transcription: ? ? Text Transcription: x=1/3 square root y(y-3), 1 less than or equal slant y less than or equal slant 9

Find the exact length of the curve. \(y=3+\frac{1}{2} \cosh 2 x\), \(0 \leqslant x \leqslant 1\)

Find the exact length of the curve. \(y=\frac{1}{4} x^{2}-\frac{1}{2} \ln x, \quad 1 \leqslant x \leqslant 2\) Equation Transcription: ?? Text Transcription: y=1/4 x^2-1/2 ln x, 1 less than or equal slant x less than or equal slant 2

Find the exact length of the curve. \(y=\sqrt{x-x^{2}}+\sin ^{-1}(\sqrt{x})\) ________________ Equation Transcription: Text Transcription: y=square root x-x^2+sin^-1(square root x)

Find the exact length of the curve. \(y=\ln \left(1-x^{2}\right), \quad 0 \leqslant x \leqslant \frac{1}{2}\) Equation Transcription: ? ? Text Transcription: y=\ln \left(1-x^{2}\right), \quad 0 \leqslant x \leqslant \frac{1}{2}

Find the exact length of the curve. \(y=1-e^{-x}, \quad 0 \leqslant x \leqslant 2\) Equation Transcription: ?? Text Transcription: y=1-e^{-x}, \quad 0 \leqslant x \leqslant 2

Find the length of the arc of the curve from point \(P\) to point \(Q\) \(y=\frac{1}{2} x^{2}, \quad P\left(-1, \frac{1}{2}\right), \quad Q\left(1, \frac{1}{2}\right)\) Equation Transcription: Text Transcription: P Q y=\frac{1}{2} x^{2}, \quad P\left(-1, \frac{1}{2}\right), \quad Q\left(1, \frac{1}{2}\right)

Find the length of the arc of the curve from point \(P\) to point \(Q\) \(x^{2}=(y-4)^{3}, \quad P(1,5), \quad Q(8,8)\) Equation Transcription: Text Transcription: P Q x^2=(y-4)^3, P(1,5), Q(8, 8)

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places. \(y=x^{2}+x^{3}, \quad 1 \leqslant x \leqslant 2\) Equation Transcription: ? ? Text Transcription: y=x^{2}+x^{3}, \quad 1 \leqslant x \leqslant 2

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places. y=x+cos x, \(0 \leqslant x \leqslant \pi / 2\)

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places. \(y=\sqrt[3]{x}, \quad 1 \leqslant x \leqslant 4\) ________________ Equation Transcription: ? ? Text Transcription: y=\sqrt[3]{x}, \quad 1 \leqslant x \leqslant 4

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places. \(y=x \tan x, \quad 0 \leqslant x \leqslant 1\) ________________ Equation Transcription: ? ? Text Transcription: y=x \tan x, \quad 0 \leqslant x \leqslant 1

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places. \(y=x e^{-x}, \quad 1 \leqslant x \leqslant 2\) Equation Transcription: ? ? Text Transcription: y=x e^{-x}, \quad 1 \leqslant x \leqslant 2

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places. \(y=\ln \left(x^2+4\right), \quad-2 \leqslant x \leqslant 2\)

Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator or computer. y = x sin x, 0 ? x ? 2????

Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator or computer. y = , 0 ? x ? 2

(a) Graph the curve y = x, 0 ? x ? 4. (b) Compute the lengths of approximating polygonal paths with n = 1, 2, and 4 segments. (Divide the interval into equal subintervals.) Illustrate by sketching the curve and these paths (as in Figure 6). (c) Set up an integral for the length of the curve. (d) Compute the length of the curve to four decimal places. Compare with the approximations in part (b).

Repeat Exercise 35 for the curve y = x + sin x 0 ? x ? 2????

Use either a computer or a table of integrals to find the exact length of the arc of the curve y = ex that lies between the points (0, 1) and (2, e2).

Use either a computer or a table of integrals to find the exact length of the arc of the curve y = x4/3 that lies between the points (0, 0) and (1, 1). If your software has trouble evaluating the integral, make a substitution that changes the integral into one that the software can evaluate.

Find the length of the astroid x2/3 + y2/3 = 1.

(a) Sketch the curve y3 = x2. (b) Use Formulas 3 and 4 to set up two integrals for the arc length from (0, 0) to (1, 1). Observe that one of these is an improper integral and evaluate both of them. (c) Find the length of the arc of this curve from (?1, 1) to (8, 4).

Find the arc length function for the curve y = 2x3/2 with starting point P0(1, 2).

(a) Find the arc length function for the curve \(y=\ln (\sin x)\), \(0<x<\pi\), with starting point \((\pi / 2,0)\). (b) Graph both the curve and its arc length function on the same screen. Why is the arc length function negative when x is less than \(\pi / 2\)?

Find the arc length function for the curve \(y=\sin ^{-1} x+\sqrt{1-x^{2}}\) with starting point \((0,1)\). ________________ Equation Transcription: Text Transcription: y=\sin ^{-1} x+\sqrt{1-x^{2}} (0, 1)

The arc length function for a curve y = f(x), where f is an increasing function, is s(x) = dt. (a) If f has y-intercept 2, find an equation for f. (b) What point on the graph of f is 3 units along the curve from the y-intercept? State your answer rounded to 3 decimal places.

A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.

A steady wind blows a kite due west. The kite's height above ground from horizontal position x=0 to \(x=80 \mathrm{ft}\) is given by \(y=150-\frac{1}{40}(x-50)^2\). Find the distance traveled by the kite.

A manufacturer of corrugated metal roofing wants to produce panels that are \(28 \mathrm{in}\). wide and \(2 \mathrm{in}\). high by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation \(y=\sin (\pi x / 7)\) and find the width w of a flat metal sheet that is needed to make a 28-inch panel. (Numerically evaluate the integral correct to four significant digits.)

Catenary Curves A chain (or cable) of uniform density that is suspended between two points, as shown in the figure, hangs in the shape of a curve called a catenary with equation y = a cosh(x/a). (See the Discovery Project following Section 12.2.) (a) Find the arc length of the catenary y = a cosh(x/a) on the interval [c, d]. (b) Show that on any interval [c, d], the ratio of the area under the catenary to its arc length is a.

The figure shows a telephone wire hanging between two poles at x=-25 and x=25. The wire hangs in the shape of a catenary described by the equation \(y=c+a \cosh \frac{x}{a}\) If the length of the wire between the two poles is \(51 \mathrm{ft}\) and the lowest point of the wire must be \(20 \mathrm{ft}\) above the ground, how high up on each pole should the wire be attached?

Catenary Curves A chain (or cable) of uniform density that is suspended between two points, as shown in the figure, hangs in the shape of a curve called a catenary with equation y = a cosh(x/a). (See the Discovery Project following Section 12.2.) The British physicist and architect Robert Hooke (1635–1703) was the first to observe that the ideal shape for a free standing arch is an inverted catenary. Hooke remarked, “As hangs the chain, so stands the arch.” The Gateway Arch in St. Louis is based on the shape of a catenary; the central curve of the arch is modeled by the equation where x and y are measured in meters and | x | ? 91.20. Set up an integral for the length of the arch and evaluate the integral numerically to estimate the length correct to the nearest meter.

For the function f(x) = ex + e?x, prove that the arc length on any interval has the same value as the area under the curve.

The curves with equations xn + yn = 1, n = 4, 6, 8, . . . , are called fat circles. Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the length L2k of the fat circle with n = 2k. Without attempting to evaluate this integral, state the value of limk L2k.

Find the length of the curve