Stewart 8th Section 8.1: Arc Length

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Stewart 8th Section 8.1: Arc Length 0/50
1 Applications of Integration · Level 1
Use the arc length formula to find the length of the curve \(y = 2 x - 5\), \(-1 \leq x \leq 3\). Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.
2 Applications of Integration · Level 2
Use the arc length formula to find the length of the curve \(y = \sqrt{2 - x^2}\), \(0 \leq x \leq 1\). Check your answer by noting that the curve is part of a circle.
3 Applications of Integration · Level 2
Set up an integral that represents the length of the curve \(y = \sin x\), \(0 \leq x \leq \pi\). Then use your calculator to find the length correct to four decimal places.
4 Applications of Integration · Level 2
Set up an integral that represents the length of the curve \(y = x e^{-x}\), \(0 \leq x \leq 2\). Then use your calculator to find the length correct to four decimal places.
5 Applications of Integration · Level 2
Set up an integral that represents the length of the curve \(y = x - \ln x\), \(1 \leq x \leq 4\). Then use your calculator to find the length correct to four decimal places.
6 Applications of Integration · Level 2
Set up an integral that represents the length of the curve \(x = y^2 - 2 y\), \(0 \leq y \leq 2\). Then use your calculator to find the length correct to four decimal places.
7 Applications of Integration · Level 2
Set up an integral that represents the length of the curve \(x = \sqrt{y} - y\), \(1 \leq y \leq 4\). Then use your calculator to find the length correct to four decimal places.
8 Applications of Integration · Level 2
Set up an integral that represents the length of the curve \(y^2 = \ln x\), \(-1 \leq y \leq 1\). Then use your calculator to find the length correct to four decimal places.
9 Applications of Integration · Level 2
Find the exact length of the curve \(y = 1 + 6 x^{\dfrac{3}{2}}\), \(0 \leq x \leq 1\).
10 Applications of Integration · Level 3
Find the exact length of the curve \(36 y^2 = (x^2 - 4)^3\), \(2 \leq x \leq 3\), \(y \geq 0\).
11 Applications of Integration · Level 2
Find the exact length of the curve \(y = \dfrac{x^3}{3} + \dfrac{1}{4 x}\), \(1 \leq x \leq 2\).
12 Applications of Integration · Level 3
Find the exact length of the curve \(x = \dfrac{y^4}{8} + \dfrac{1}{4 y^2}\), \(1 \leq y \leq 2\).
13 Applications of Integration · Level 3
Find the exact length of the curve \(x = \dfrac{1}{3} \sqrt{y}(y - 3)\), \(1 \leq y \leq 9\).
14 Applications of Integration · Level 3
Find the exact length of the curve \(y = \ln(\cos x)\), \(0 \leq x \leq \dfrac{\pi}{3}\).
15 Applications of Integration · Level 3
Find the exact length of the curve \(y = \ln(\sec x)\), \(0 \leq x \leq \dfrac{\pi}{4}\).
16 Applications of Integration · Level 2
Find the exact length of the curve \(y = 3 + \dfrac{1}{2} \cosh 2 x\), \(0 \leq x \leq 1\).
17 Applications of Integration · Level 3
Find the exact length of the curve \(y = \dfrac{1}{4} x^2 - \dfrac{1}{2} \ln x\), \(1 \leq x \leq 2\).
18 Applications of Integration · Level 4
Find the exact length of the curve \(y = \sqrt{x - x^2} + \sin^{-1}(\sqrt{x})\).
19 Applications of Integration · Level 3
Find the exact length of the curve \(y = \ln(1 - x^2)\), \(0 \leq x \leq \dfrac{1}{2}\).
20 Applications of Integration · Level 3
Find the exact length of the curve \(y = 1 - e^{-x}\), \(0 \leq x \leq 2\).
21 Applications of Integration · Level 2
Find the length of the arc of the curve from point \(P\) to point \(Q\): \(y = \dfrac{1}{2} x^2\), \(P\left(-1, \dfrac{1}{2}\right)\), \(Q\left(1, \dfrac{1}{2}\right)\).
22 Applications of Integration · Level 3
Find the length of the arc of the curve from point \(P\) to point \(Q\): \(x^2 = (y - 4)^3\), \(P(1, 5)\), \(Q(8, 8)\).
23 Applications of Integration · Level 2
Graph the curve \(y = x^2 + x^3\), \(1 \leq x \leq 2\) and visually estimate its length. Then use your calculator to find the length correct to four decimal places.
24 Applications of Integration · Level 2
Graph the curve \(y = x + \cos x\), \(0 \leq x \leq \dfrac{\pi}{2}\) and visually estimate its length. Then use your calculator to find the length correct to four decimal places.
25 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to estimate the arc length of the curve \(y = x \sin x\), \(0 \leq x \leq 2 \pi\). Compare your answer with the value of the integral produced by a calculator.
26 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to estimate the arc length of the curve \(y = \sqrt[3]{x}\), \(1 \leq x \leq 6\). Compare your answer with the value of the integral produced by a calculator.
27 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to estimate the arc length of the curve \(y = \ln(1 + x^3)\), \(0 \leq x \leq 5\). Compare your answer with the value of the integral produced by a calculator.
28 Applications of Integration · Level 3
Use Simpson's Rule with \(n = 10\) to estimate the arc length of the curve \(y = e^{-x^2}\), \(0 \leq x \leq 2\). Compare your answer with the value of the integral produced by a calculator.
29 Applications of Integration · Level 4
(a) Graph the curve \(y = x \sqrt[3]{4 - x}\), \(0 \leq x \leq 4\). (b) Compute the lengths of inscribed polygons with \(n = 1\), \(2\), and \(4\) sides (using equal subintervals). Illustrate by sketching these polygons. (c) Set up an integral for the length of the curve. (d) Use your calculator to find the length of the curve to four decimal places. Compare with the approximations in part (b).
30 Applications of Integration · Level 4
Repeat Exercise 29 for the curve \(y = x + \sin x\), \(0 \leq x \leq 2 \pi\).
31 Applications of Integration · Level 3
Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve \(y = e^x\) that lies between the points \((0, 1)\) and \((2, e^2)\).
32 Applications of Integration · Level 3
Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve \(y = x^{\dfrac{4}{3}}\) that lies between the points \((0, 0)\) and \((1, 1)\). If your CAS has trouble evaluating the integral, make a substitution that changes the integral into one that the CAS can evaluate.
33 Applications of Integration · Level 3
Sketch the curve with equation \(x^{\dfrac{2}{3}} + y^{\dfrac{2}{3}} = 1\) and use symmetry to find its length.
34 Applications of Integration · Level 4
(a) Sketch the curve \(y^3 = x^2\). (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 \((8, 4)\).
35 Applications of Integration · Level 2
Find the arc length function for the curve \(y = 2 x^{\dfrac{3}{2}}\) with starting point \(P_0(1, 2)\).
36 Applications of Integration · Level 3
(a) Find the arc length function for the curve \(y = \ln(\sin x)\), \(0 < x < \pi\), with starting point \(\left(\dfrac{\pi}{2}, 0\right)\). (b) Graph both the curve and its arc length function on the same screen.
37 Applications of Integration · Level 3
Find the arc length function for the curve \(y = \sin^{-1} x + \sqrt{1 - x^2}\) with starting point \((0, 1)\).
38 Applications of Integration · Level 4
The arc length function for a curve \(y = f(x)\), where \(f\) is an increasing function, is \(s(x) = \displaystyle\int_{0}^{x} \sqrt{3 t + 5} d t\). (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.
39 Applications of Integration · Level 3
For the function \(f(x) = \dfrac{1}{4} e^x + e^{-x}\), prove that the arc length on any interval has the same value as the area under the curve.
40 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
A steady wind blows a kite due west. The kite's height above ground from horizontal position \(x = 0\) to \(x = 80\) ft is given by \(y = 150 - \dfrac{1}{40}(x - 50)^2\). Find the distance traveled by the kite.
41 Applications of Integration · Level 3
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 \(y = 180 - \dfrac{x^2}{45}\) 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.
42 Applications of Integration · Level 3
The Gateway Arch in St. Louis was constructed using the equation \(y = 211.49 - 20.96 \cosh 0.03291765 x\) for the central curve of the arch, where \(x\) and \(y\) are measured in meters and \(|x| \leq 91.20\). Set up an integral for the length of the arch and use your calculator to estimate the length correct to the nearest meter.
43 Applications of Integration · Level 3
A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. high by processing flat sheets of metal. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation \(y = \sin\left(\pi \dfrac{x}{7}\right)\) and find the width \(w\) of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)
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44 Applications of Integration · Level 4
(a) The figure shows a telephone wire hanging between two poles at \(x = -b\) and \(x = b\). It takes the shape of a catenary with equation \(y = c + a \cosh\left(\dfrac{x}{a}\right)\). Find the length of the wire. (b) Suppose two telephone poles are 50 ft apart and the length of the wire between the poles is 51 ft. If the lowest point of the wire must be 20 ft above the ground, how high up on each pole should the wire be attached?
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45 Applications of Integration · Level 3
Find the length of the curve \(y = \displaystyle\int_{1}^{x} \sqrt{t^3 - 1} d t\), \(1 \leq x \leq 4\).
46 Applications of Integration · Level 4
The curves with equations \(x^n + y^n = 1\), \(n = 4, 6, 8, \ldots\), 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 \(L_{2 k}\) of the fat circle with \(n = 2 k\). Without attempting to evaluate this integral, state the value of \(\operatorname*{lim}\limits_{k \rightarrow \infty} L_{2 k}\).
47 Applications of Integration · Level 2
Find the length of the arc of the semicubical parabola \(y^2 = x^3\) between the points \((1, 1)\) and \((4, 8)\).
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48 Applications of Integration · Level 3
Find the length of the arc of the parabola \(y^2 = x\) from \((0, 0)\) to \((1, 1)\).
49 Applications of Integration · Level 3
(a) Set up an integral for the length of the arc of the hyperbola \(x y = 1\) from the point \((1, 1)\) to the point \(\left(2, \dfrac{1}{2}\right)\). (b) Use Simpson's Rule with \(n = 10\) to estimate the arc length.
50 Applications of Integration · Level 3
Find the arc length function for the curve \(y = x^2 - \dfrac{1}{8} \ln x\) taking \(P_0(1, 1)\) as the starting point.
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