Explanation Formula used: The length of the curve is L = ∫ 0 π r 2 + ( d r d θ ) 2 d θ where the function r = f ( θ ) . Calculation: Consider the polar equation r = sin 3 θ 3 . Differentiate the equation r = sin 3 θ 3 with respect to θ . d r d θ = [ 3 sin 2 ( θ 3 ) ] cos ( θ 3 ) ( 1 3 ) d r d θ = sin 2 ( θ 3 ) cos ( θ 3 ) Length is determined for the limits of 0 to π . Use above mentioned formula and find the length of curve as follows. L = ∫ 0 π r 2 + ( d r d θ ) 2 d θ = ∫ 0 π ( sin 3 θ 3 ) 2 + ( sin 2 ( θ 3 ) cos ( θ 3 ) ) 2 d θ = ∫ 0 π sin 6 θ 3 + sin 4 ( θ 3 ) cos 2 ( θ 3 ) d θ = ∫ 0 π sin 4 θ 3 [ sin 2 ( θ 3 ) + cos 2 ( θ 3 ) ] d θ On further simplification,

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 9, Problem 40RE

To determine

The length of the curve r=sin3θ3 for 0≤θ≤π.

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Chapter 9 Solutions

Essential Calculus: Early Transcendentals

Ch. 9.1 - 14 Sketch the curve by using the parametric...Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - (a) Eliminate the parameter to find a Cartesian...Ch. 9.1 - (a) Eliminate the parameter to find a Cartesian...

Ch. 9.1 - (a) Eliminate the parameter to find a Cartesian...Ch. 9.1 - (a) Eliminate the parameter to find a Cartesian...Ch. 9.1 - Prob. 13E Ch. 9.1 - Prob. 14E Ch. 9.1 - 15-18 Describe the motion of a particle with...Ch. 9.1 - 15-18 Describe the motion of a particle with...Ch. 9.1 - Describe the motion of a particle with position...Ch. 9.1 - Describe the motion of a particle with position...Ch. 9.1 - Prob. 19E Ch. 9.1 - Prob. 20E Ch. 9.1 - Prob. 21E Ch. 9.1 - Match the parametric equations with the graphs...Ch. 9.1 - Graph the curve x = y 2 sin y.Ch. 9.1 - Graph the curves y = x3 4x and x = y3 4y and...Ch. 9.1 - (a) Show that the parametric equations x = x1 +...Ch. 9.1 - Use a graphing device and the result of Exercise...Ch. 9.1 - Find parametric equations for the path of a...Ch. 9.1 - (a) Find parametric equations for the ellipse...Ch. 9.1 - Use a graphing calculator or computer to reproduce...Ch. 9.1 - Use a graphing calculator or computer to reproduce...Ch. 9.1 - Compare the curves represented by the parametric...Ch. 9.1 - Prob. 32E Ch. 9.1 - Derive Equations 1 for the case /2 Ch. 9.1 - Let P be a point at a distance d from the center...Ch. 9.1 - If a and b are fixed numbers, find parametric...Ch. 9.1 - A curve, called a witch of Maria Agnesi, consists...Ch. 9.1 - Prob. 37E Ch. 9.1 - Prob. 38E Ch. 9.1 - Investigate the family of curves defined by the...Ch. 9.1 - The swallowtail catastrophe curves are defined by...Ch. 9.1 - Graph several members of the family of curves with...Ch. 9.1 - Graph several members of the family of curves x =...Ch. 9.1 - The curves with x = a sin nt, y = b cos t, called...Ch. 9.1 - Prob. 44E Ch. 9.2 - Find dy/dx. x = t sin t, y = t2 + t Ch. 9.2 - Find dy/dx. x = 1/t, y=tet Ch. 9.2 - Find an equation of the tangent to the curve at...Ch. 9.2 - Find an equation of the tangent to the curve at...Ch. 9.2 - Find an equation of the tangent to the curve at...Ch. 9.2 - Find an equation of the tangent to the curve at...Ch. 9.2 - Find an equation of the tangent to the curve at...Ch. 9.2 - Find equations of the tangents to the curve x =...Ch. 9.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 9.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 9.2 - Find dy/dx and d2y/dx2. For which values of t is...Ch. 9.2 - 912 Find dy/dx and d2y/dx2. For which values t of...Ch. 9.2 - Prob. 13E Ch. 9.2 - Prob. 14E Ch. 9.2 - Find the points on the curve where the tangent is...Ch. 9.2 - Find the points on the curve where the tangent is...Ch. 9.2 - Prob. 17E Ch. 9.2 - Prob. 18E Ch. 9.2 - Prob. 19E Ch. 9.2 - Prob. 20E Ch. 9.2 - Show that the curve x = cos t, y = sin t cos t has...Ch. 9.2 - Graph the curve x = cos t + 2 cos 2t, y = sin t +...Ch. 9.2 - Prob. 23E Ch. 9.2 - Prob. 24E Ch. 9.2 - At what points on the curve x = 2t3, y = 1 + 4t ...Ch. 9.2 - Prob. 26E Ch. 9.2 - Prob. 27E Ch. 9.2 - Find the area enclosed by the curve x = t2 2t,...Ch. 9.2 - Find the area enclosed by the x-axis and the curve...Ch. 9.2 - Prob. 30E Ch. 9.2 - Prob. 31E Ch. 9.2 - Prob. 32E Ch. 9.2 - Prob. 33E Ch. 9.2 - Prob. 34E Ch. 9.2 - Prob. 35E Ch. 9.2 - Prob. 36E Ch. 9.2 - Prob. 37E Ch. 9.2 - Prob. 38E Ch. 9.2 - Prob. 39E Ch. 9.2 - Prob. 40E Ch. 9.2 - Prob. 41E Ch. 9.2 - Prob. 42E Ch. 9.2 - Prob. 43E Ch. 9.2 - Find the length of the loop of the curve x = 3t ...Ch. 9.2 - Prob. 45E Ch. 9.2 - Prob. 46E Ch. 9.2 - Prob. 47E Ch. 9.2 - Prob. 48E Ch. 9.2 - Prob. 49E Ch. 9.2 - Prob. 50E Ch. 9.2 - Prob. 51E Ch. 9.2 - A string is wound around a circle and then unwound...Ch. 9.2 - A cow is tied to a silo with radius r by a rope...Ch. 9.3 - Prob. 1E Ch. 9.3 - Prob. 2E Ch. 9.3 - Prob. 3E Ch. 9.3 - Prob. 4E Ch. 9.3 - Prob. 5E Ch. 9.3 - Prob. 6E Ch. 9.3 - Prob. 7E Ch. 9.3 - Prob. 8E Ch. 9.3 - Prob. 9E Ch. 9.3 - Prob. 10E Ch. 9.3 - Prob. 11E Ch. 9.3 - Prob. 12E Ch. 9.3 - Identify the curve by finding a Cartesian equation...Ch. 9.3 - Identify the curve by finding a Cartesian equation...Ch. 9.3 - Identify the curve by finding a Cartesian equation...Ch. 9.3 - Prob. 16E Ch. 9.3 - Prob. 17E Ch. 9.3 - Prob. 18E Ch. 9.3 - Prob. 19E Ch. 9.3 - Prob. 20E Ch. 9.3 - Prob. 21E Ch. 9.3 - Prob. 22E Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - 2340 Sketch the curve with the given polar...Ch. 9.3 - 2340 Sketch the curve with the given polar...Ch. 9.3 - Prob. 30E Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - 2340 Sketch the curve with the given polar...Ch. 9.3 - Prob. 34E Ch. 9.3 - Prob. 35E Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - Sketch the curve with the given polar equation by...Ch. 9.3 - Prob. 38E Ch. 9.3 - Prob. 39E Ch. 9.3 - Prob. 40E Ch. 9.3 - Prob. 41E Ch. 9.3 - Prob. 42E Ch. 9.3 - Prob. 43E Ch. 9.3 - Prob. 45E Ch. 9.3 - Prob. 44E Ch. 9.3 - Prob. 46E Ch. 9.3 - Prob. 47E Ch. 9.3 - Find the slope of the tangent line to the given...Ch. 9.3 - Prob. 49E Ch. 9.3 - Find the slope of the tangent line to the given...Ch. 9.3 - Prob. 51E Ch. 9.3 - Prob. 54E Ch. 9.3 - Prob. 53E Ch. 9.3 - Find the points on the given curve where the...Ch. 9.3 - Prob. 55E Ch. 9.3 - Prob. 56E Ch. 9.3 - Prob. 57E Ch. 9.3 - Prob. 60E Ch. 9.3 - Prob. 59E Ch. 9.3 - Prob. 58E Ch. 9.3 - How are the graphs of r = 1 + sin( /6) and r = 1...Ch. 9.3 - Use a graph to estimate the y-coordinate of the...Ch. 9.3 - Prob. 63E Ch. 9.3 - Prob. 64E Ch. 9.3 - A family of curves has polar equations...Ch. 9.3 - Let P be any point (except the origin) on the...Ch. 9.3 - Prob. 68E Ch. 9.4 - Find the area of the region that is bounded by the...Ch. 9.4 - Prob. 2E Ch. 9.4 - Find the area of the region that is bounded by the...Ch. 9.4 - Prob. 4E Ch. 9.4 - Prob. 6E Ch. 9.4 - Prob. 8E Ch. 9.4 - Find the area of the shaded region. 7.Ch. 9.4 - Find the area of the shaded region. 5.Ch. 9.4 - Sketch the curve and find the area that it...Ch. 9.4 - Sketch the curve and find the area that it...Ch. 9.4 - Prob. 11E Ch. 9.4 - 912 Sketch the curve and find the area that it...Ch. 9.4 - Prob. 14E Ch. 9.4 - Graph the curve and find the area that it...Ch. 9.4 - Find the area of the region enclosed by one loop...Ch. 9.4 - 1518 Find the area of the region enclosed by one...Ch. 9.4 - Prob. 17E Ch. 9.4 - Prob. 18E Ch. 9.4 - 19-22 Find the area of the region that lies inside...Ch. 9.4 - Find the area of the region that lies inside the...Ch. 9.4 - Find the area of the region that lies inside the...Ch. 9.4 - Prob. 22E Ch. 9.4 - 2326 Find the area of the region that lies inside...Ch. 9.4 - Find the area of the region that lies inside both...Ch. 9.4 - Find the area of the region that lies inside both...Ch. 9.4 - Prob. 26E Ch. 9.4 - Prob. 27E Ch. 9.4 - Prob. 29E Ch. 9.4 - Prob. 30E Ch. 9.4 - Find all points of intersection of the given...Ch. 9.4 - Prob. 32E Ch. 9.4 - When recording live performances, sound engineers...Ch. 9.4 - 3336 Find the exact length of the polar curve. 33....Ch. 9.4 - 3336 Find the exact length of the polar curve. 34....Ch. 9.4 - Find the exact length of the polar curve. 47. r =...Ch. 9.4 - Prob. 36E Ch. 9.4 - Prob. 37E Ch. 9.4 - Prob. 38E Ch. 9.5 - Prob. 1E Ch. 9.5 - Prob. 2E Ch. 9.5 - Prob. 3E Ch. 9.5 - Prob. 4E Ch. 9.5 - Prob. 7E Ch. 9.5 - Prob. 6E Ch. 9.5 - Prob. 5E Ch. 9.5 - Prob. 8E Ch. 9.5 - Prob. 9E Ch. 9.5 - Prob. 10E Ch. 9.5 - Prob. 11E Ch. 9.5 - Prob. 12E Ch. 9.5 - Prob. 13E Ch. 9.5 - Prob. 14E Ch. 9.5 - Prob. 15E Ch. 9.5 - Prob. 16E Ch. 9.5 - Prob. 17E Ch. 9.5 - Prob. 18E Ch. 9.5 - Prob. 19E Ch. 9.5 - Prob. 20E Ch. 9.5 - Prob. 21E Ch. 9.5 - Prob. 22E Ch. 9.5 - Prob. 23E Ch. 9.5 - Prob. 24E Ch. 9.5 - The orbit of Halleys comet, last seen in 1986 and...Ch. 9.5 - Prob. 26E Ch. 9.5 - Prob. 27E Ch. 9.5 - The distance from the dwarf planet Pluto to the...Ch. 9.5 - Prob. 29E Ch. 9 - Prob. 1RCC Ch. 9 - Prob. 2RCC Ch. 9 - Prob. 3RCC Ch. 9 - Prob. 4RCC Ch. 9 - Prob. 5RCC Ch. 9 - Prob. 6RCC Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Prob. 2RQ Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Prob. 4RQ Ch. 9 - Prob. 5RQ Ch. 9 - Prob. 6RQ Ch. 9 - Prob. 7RQ Ch. 9 - Prob. 8RQ Ch. 9 - Prob. 1RE Ch. 9 - Prob. 2RE Ch. 9 - Prob. 3RE Ch. 9 - Prob. 4RE Ch. 9 - Prob. 5RE Ch. 9 - Prob. 6RE Ch. 9 - Prob. 7RE Ch. 9 - Sketch the region consisting of points whose polar...Ch. 9 - Prob. 9RE Ch. 9 - Prob. 10RE Ch. 9 - Sketch the polar curve. 11. r = cos 3 Ch. 9 - Sketch the polar curve. 12. r = 3 + cos 3 Ch. 9 - Sketch the polar curve. 13. r = 1 + cos 2 Ch. 9 - Sketch the polar curve. 14. r = 2 cos(/2)Ch. 9 - Sketch the polar curve. 15. r=31+2sin Ch. 9 - Sketch the polar curve. 16. r=322cos Ch. 9 - Find a polar equation for the curve represented by...Ch. 9 - Find a polar equation for the curve represented by...Ch. 9 - The curve with polar equation r = (sin )/ is...Ch. 9 - Prob. 20RE Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find dy/dx and d2y/dx2. 25. x = t + sin t, y = t ...Ch. 9 - Prob. 26RE Ch. 9 - Prob. 27RE Ch. 9 - Prob. 28RE Ch. 9 - At what points does the curve x = 2a cos t a cos...Ch. 9 - Find the area enclosed by the curve in Exercise...Ch. 9 - Prob. 31RE Ch. 9 - Prob. 32RE Ch. 9 - Prob. 33RE Ch. 9 - Prob. 34RE Ch. 9 - Prob. 35RE Ch. 9 - Prob. 36RE Ch. 9 - Prob. 37RE Ch. 9 - Prob. 38RE Ch. 9 - Prob. 39RE Ch. 9 - Prob. 40RE Ch. 9 - The curves defined by the parametric equations...Ch. 9 - Prob. 42RE Ch. 9 - Prob. 43RE Ch. 9 - Prob. 44RE Ch. 9 - In the figure the circle of radius a is...

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The length of the curve r = sin 3 θ 3 for 0 ≤ θ ≤ π .

Solution Summary: The author calculates the length of the curve r=mathrmsin3theta.

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The length of the curve r=sin3θ3 for 0≤θ≤π.

Chapter 9 Solutions