Explanation Given: The parametric equation for the variable x is as below. x = cos t + ln ( tan 1 2 t ) The parametric equation for the variable y is as below. y = sin t The value t ranges from π 4 to 3 π 4 . Calculation: The length of the curve is obtained by the formula. L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t Differentiate the variable x with respect to t . x = cos t + ln ( tan 1 2 t ) d x d t = − sin t + 1 2 sec 2 ( t 2 ) tan ( t 2 ) = − sin t + 1 2 sin ( t 2 ) cos ( t 2 ) = − sin t + 1 sin t Differentiate the variable y with respect to t . y = sin t d y d t = cos t Write the length of the curve formula as below. L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t Substitute ( − sin t + 1 sin t ) for d x d t , ( cos t ) for d y d t , π 4 for α , and 3 π 4 for β in the above equation, L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t = ∫ π 4 3 π 4 ( − sin t + 1 sin t ) 2 + ( cos t ) 2 d t = 2 ∫ π 4 π 2 ( − sin t + 1 sin t ) 2 + ( cos t ) 2 d t = 2 ∫ π 4 π 2 − sin 2 t + 1 sin 2 t + cos t d t Simplify further, L = 2 ∫ π 4 π 2 cot t d t = 2 ∫ π 4 π 2 1 − 2 + 1 sin 2 t d t = 2 ∫ π 4 π 2 csc 2<

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Chapter 10.2, Problem 46E

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To find: The length of the curve and graph the curve obtained from the parametric equations x=cost+ln(tan12t) and y=sint.

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Chapter 10.2, Problem 45E

Chapter 10.2, Problem 47E

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Bundle: Single Variable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Multi-Term

Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Describe the motion of a particle with position...Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Match the parametric equations with the graphs...Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Use a graphing device and the result of Exercise...Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.1 - Prob. 41E Ch. 10.1 - Prob. 42E Ch. 10.1 - Prob. 43E Ch. 10.1 - (a) Find parametric equations for the set of all...Ch. 10.1 - Suppose that the position of one particle at time...Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Prob. 18E Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Use a graph to estimate the coordinates of the...Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.2 - (a) Find the slope of the tangent line to the...Ch. 10.2 - Prob. 28E Ch. 10.2 - At what point(s) on the curve x = 3t2 + 1, y = t3 ...Ch. 10.2 - Prob. 30E Ch. 10.2 - Use the parametric equations of an ellipse, x = a...Ch. 10.2 - Find the area enclosed by the curve x = t2 2t,...Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - Prob. 35E Ch. 10.2 - Prob. 36E Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Find the exact length of the curve. 41. x = 1 +...Ch. 10.2 - Find the exact length of the curve. 42. x = et t,...Ch. 10.2 - Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.2 - Prob. 45E Ch. 10.2 - Prob. 46E Ch. 10.2 - Prob. 47E Ch. 10.2 - Prob. 48E Ch. 10.2 - Prob. 49E Ch. 10.2 - In Exercise 10.1.43 you were asked to derive the...Ch. 10.2 - Prob. 51E Ch. 10.2 - Prob. 52E Ch. 10.2 - Prob. 53E Ch. 10.2 - Prob. 54E Ch. 10.2 - (a) Graph the epitrochoid with equations...Ch. 10.2 - Prob. 57E Ch. 10.2 - Set up an integral that represents the area of the...Ch. 10.2 - Prob. 59E Ch. 10.2 - Prob. 60E Ch. 10.2 - Prob. 61E Ch. 10.2 - Prob. 62E Ch. 10.2 - Prob. 63E Ch. 10.2 - Prob. 64E Ch. 10.2 - Prob. 65E Ch. 10.2 - Prob. 66E Ch. 10.2 - Prob. 67E Ch. 10.2 - Prob. 68E Ch. 10.2 - Prob. 69E Ch. 10.2 - (a) Use the formula in Exercise 69(b) to find the...Ch. 10.2 - Prob. 71E Ch. 10.2 - Prob. 72E Ch. 10.2 - A string is wound around a circle and then unwound...Ch. 10.2 - A cow is tied to a silo with radius r by a rope...Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Plot the point whose polar coordinates are given....Ch. 10.3 - Plot the point whose polar coordinates are given....Ch. 10.3 - The Cartesian coordinates of a point are given....Ch. 10.3 - Prob. 6E Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Prob. 10E Ch. 10.3 - Prob. 11E Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Identify the curve by finding a Cartesian equation...Ch. 10.3 - Identify the curve by finding a Cartesian equation...Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 36E Ch. 10.3 - Prob. 37E Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.3 - Prob. 40E Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Sketch the curve with the given polar equation by...Ch. 10.3 - Prob. 43E Ch. 10.3 - Prob. 44E Ch. 10.3 - Prob. 45E Ch. 10.3 - Prob. 46E Ch. 10.3 - Prob. 47E Ch. 10.3 - Prob. 48E Ch. 10.3 - Prob. 49E Ch. 10.3 - Prob. 50E Ch. 10.3 - Prob. 51E Ch. 10.3 - Prob. 52E Ch. 10.3 - Prob. 53E Ch. 10.3 - Prob. 54E Ch. 10.3 - Prob. 55E Ch. 10.3 - Prob. 56E Ch. 10.3 - Prob. 57E Ch. 10.3 - Prob. 58E Ch. 10.3 - Prob. 59E Ch. 10.3 - Prob. 60E Ch. 10.3 - Prob. 61E Ch. 10.3 - Prob. 62E Ch. 10.3 - Prob. 63E Ch. 10.3 - Prob. 64E Ch. 10.3 - Prob. 65E Ch. 10.3 - Prob. 66E Ch. 10.3 - Use a graphing device to graph the polar curve....Ch. 10.3 - Prob. 68E Ch. 10.3 - Use a graphing device to graph the polar curve....Ch. 10.3 - Prob. 70E Ch. 10.3 - Use a graphing device to graph the polar curve....Ch. 10.3 - Prob. 72E Ch. 10.3 - Prob. 73E Ch. 10.3 - Prob. 74E Ch. 10.3 - Prob. 75E Ch. 10.3 - Prob. 76E Ch. 10.3 - Prob. 77E Ch. 10.3 - Prob. 78E Ch. 10.4 - Find the area of the region that is bounded by the...Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Find the area of the shaded region. 5.Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Prob. 10E Ch. 10.4 - Prob. 11E Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10.4 - Prob. 23E Ch. 10.4 - Prob. 24E Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Prob. 26E Ch. 10.4 - Prob. 27E Ch. 10.4 - Prob. 28E Ch. 10.4 - Prob. 29E Ch. 10.4 - Prob. 30E Ch. 10.4 - Prob. 31E Ch. 10.4 - Prob. 32E Ch. 10.4 - Prob. 33E Ch. 10.4 - Prob. 34E Ch. 10.4 - Prob. 35E Ch. 10.4 - Prob. 36E Ch. 10.4 - Prob. 37E Ch. 10.4 - Prob. 38E Ch. 10.4 - Prob. 39E Ch. 10.4 - Prob. 40E Ch. 10.4 - Prob. 41E Ch. 10.4 - Prob. 42E Ch. 10.4 - The points of intersection of the cardioid r = 1 +...Ch. 10.4 - When recording live performances, sound engineers...Ch. 10.4 - Prob. 45E Ch. 10.4 - Find the exact length of the polar curve. 46. r =...Ch. 10.4 - Prob. 47E Ch. 10.4 - Prob. 48E Ch. 10.4 - Find the exact length of the curve. Use a graph to...Ch. 10.4 - Find the exact length of the curve. Use a graph to...Ch. 10.4 - Prob. 51E Ch. 10.4 - Prob. 52E Ch. 10.4 - Prob. 53E Ch. 10.4 - Use a calculator to find the length of the curve...Ch. 10.4 - (a) Use Formula 10.2.6 to show that the area of...Ch. 10.4 - Prob. 56E Ch. 10.5 - Prob. 1E Ch. 10.5 - Prob. 2E Ch. 10.5 - Prob. 3E Ch. 10.5 - Prob. 4E Ch. 10.5 - Prob. 5E Ch. 10.5 - Prob. 6E Ch. 10.5 - Prob. 7E Ch. 10.5 - Prob. 8E Ch. 10.5 - Prob. 9E Ch. 10.5 - Prob. 10E Ch. 10.5 - Prob. 11E Ch. 10.5 - Prob. 12E Ch. 10.5 - Prob. 13E Ch. 10.5 - Prob. 14E Ch. 10.5 - Prob. 15E Ch. 10.5 - Prob. 16E Ch. 10.5 - Prob. 17E Ch. 10.5 - Prob. 18E Ch. 10.5 - Prob. 19E Ch. 10.5 - Prob. 20E Ch. 10.5 - Prob. 21E Ch. 10.5 - Prob. 22E Ch. 10.5 - Prob. 23E Ch. 10.5 - Prob. 24E Ch. 10.5 - Prob. 25E Ch. 10.5 - Prob. 26E Ch. 10.5 - Prob. 27E Ch. 10.5 - Prob. 28E Ch. 10.5 - Prob. 29E Ch. 10.5 - Prob. 30E Ch. 10.5 - Prob. 31E Ch. 10.5 - Prob. 32E Ch. 10.5 - Prob. 33E Ch. 10.5 - Prob. 34E Ch. 10.5 - Prob. 35E Ch. 10.5 - Prob. 36E Ch. 10.5 - Prob. 37E Ch. 10.5 - Prob. 38E Ch. 10.5 - Prob. 39E Ch. 10.5 - Prob. 40E Ch. 10.5 - Prob. 41E Ch. 10.5 - Prob. 42E Ch. 10.5 - Prob. 43E Ch. 10.5 - Prob. 44E Ch. 10.5 - Prob. 45E Ch. 10.5 - Prob. 46E Ch. 10.5 - Prob. 47E Ch. 10.5 - Prob. 48E Ch. 10.5 - Prob. 49E Ch. 10.5 - Prob. 50E Ch. 10.5 - Prob. 51E Ch. 10.5 - Prob. 52E Ch. 10.5 - Show that the function defined by the upper branch...Ch. 10.5 - Prob. 54E Ch. 10.5 - Determine the type of curve represented by the...Ch. 10.5 - Prob. 56E Ch. 10.5 - Show that the tangent lines to the parabola x2 =...Ch. 10.5 - Show that if an ellipse and a hyperbola have the...Ch. 10.5 - Prob. 59E Ch. 10.5 - Prob. 60E Ch. 10.5 - Find the area of the region enclosed by the...Ch. 10.5 - Prob. 62E Ch. 10.5 - Prob. 63E Ch. 10.5 - (a) Calculate the surface area of the ellipsoid...Ch. 10.5 - Prob. 65E Ch. 10.5 - Prob. 66E Ch. 10.6 - Prob. 1E Ch. 10.6 - Prob. 2E Ch. 10.6 - Prob. 3E Ch. 10.6 - Prob. 4E Ch. 10.6 - Prob. 5E Ch. 10.6 - Prob. 6E Ch. 10.6 - Prob. 7E Ch. 10.6 - Prob. 8E Ch. 10.6 - Prob. 9E Ch. 10.6 - Prob. 10E Ch. 10.6 - Prob. 11E Ch. 10.6 - Prob. 12E Ch. 10.6 - Prob. 13E Ch. 10.6 - Prob. 14E Ch. 10.6 - Prob. 15E Ch. 10.6 - Prob. 16E Ch. 10.6 - Prob. 17E Ch. 10.6 - Prob. 18E Ch. 10.6 - Prob. 19E Ch. 10.6 - Prob. 20E Ch. 10.6 - Prob. 21E Ch. 10.6 - Prob. 22E Ch. 10.6 - Prob. 23E Ch. 10.6 - Prob. 24E Ch. 10.6 - Prob. 25E Ch. 10.6 - Prob. 26E Ch. 10.6 - The orbit of Halleys comet, last seen in 1986 and...Ch. 10.6 - Prob. 28E Ch. 10.6 - The planet Mercury travels in an elliptical orbit...Ch. 10.6 - The distance from the dwarf planet Pluto to the...Ch. 10.6 - Prob. 31E Ch. 10 - Prob. 1RCC Ch. 10 - Prob. 2RCC Ch. 10 - Prob. 3RCC Ch. 10 - Prob. 4RCC Ch. 10 - Prob. 5RCC Ch. 10 - Prob. 6RCC Ch. 10 - Prob. 7RCC Ch. 10 - Prob. 8RCC Ch. 10 - Prob. 9RCC Ch. 10 - Prob. 1RQ Ch. 10 - Prob. 2RQ Ch. 10 - Prob. 3RQ Ch. 10 - Prob. 4RQ Ch. 10 - Determine whether the statement is true or false....Ch. 10 - Prob. 6RQ Ch. 10 - Prob. 7RQ Ch. 10 - Prob. 8RQ Ch. 10 - Prob. 9RQ Ch. 10 - Prob. 10RQ Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Prob. 12RE Ch. 10 - Prob. 13RE Ch. 10 - Sketch the polar curve. 14. r = 2 cos(/2)Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE Ch. 10 - Prob. 21RE Ch. 10 - Prob. 22RE Ch. 10 - Prob. 23RE Ch. 10 - Prob. 24RE Ch. 10 - Prob. 25RE Ch. 10 - Prob. 26RE Ch. 10 - Prob. 27RE Ch. 10 - Prob. 28RE Ch. 10 - Prob. 29RE Ch. 10 - Prob. 30RE Ch. 10 - Prob. 31RE Ch. 10 - Prob. 32RE Ch. 10 - Find the points of intersection of the curves r =...Ch. 10 - Prob. 34RE Ch. 10 - Find the area of the region that lies inside both...Ch. 10 - Find the area of the region that lies inside the...Ch. 10 - Prob. 37RE Ch. 10 - Prob. 38RE Ch. 10 - Prob. 39RE Ch. 10 - Prob. 40RE Ch. 10 - Find the area of the surface obtained by rotating...Ch. 10 - Prob. 42RE Ch. 10 - Prob. 43RE Ch. 10 - Prob. 44RE Ch. 10 - Prob. 45RE Ch. 10 - Prob. 46RE Ch. 10 - Prob. 47RE Ch. 10 - Prob. 48RE Ch. 10 - Prob. 49RE Ch. 10 - Prob. 50RE Ch. 10 - Prob. 51RE Ch. 10 - Prob. 52RE Ch. 10 - Prob. 53RE Ch. 10 - Prob. 54RE Ch. 10 - Prob. 55RE Ch. 10 - Prob. 56RE Ch. 10 - Prob. 57RE Ch. 10 - A curve called the folium of Descartes is defined...Ch. 10 - Prob. 1P Ch. 10 - Prob. 2P Ch. 10 - What is the smallest viewing rectangle that...Ch. 10 - Four bugs are placed at the four comers of a...Ch. 10 - Prob. 5P Ch. 10 - A circle C of radius 2r has its center at the...

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The length of the curve and graph the curve obtained from the parametric equations x = cos t + ln ( tan 1 2 t ) and y = sin t .

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To find: The length of the curve and graph the curve obtained from the parametric equations x=cost+ln(tan12t) and y=sint.

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Chapter 10 Solutions