Explanation Equate r = 1 2 + cos θ to 0 as, 1 2 + cos θ = 0 cos θ = − 1 2 θ = cos − 1 ( − 1 2 ) θ = 2 π 3 Use online graphing calculator and sketch the graph of r = 1 2 + cos θ as shown below in Figure 1. From Figure 1, the darker shaded region (from θ = 0 to θ = 2 π 3 ) represent half of the desired area which should be added with upper half area of the inner loop. From this area, subtract half the area of the lower part of the inner loop (the lighter shaded region from θ = 2 π 3 to θ = π ) and then double that difference to obtain the desired area. Therefore, integrate the top half of the larger loop from 0 to 2 π 3 and the bottom half of the small loop from 2 π 3 to π . Calculate the area of the region using the polar area formula. A = 2 [ ( 1 2 ∫ a b r 2 d θ ) b i g l o o p − ( 1 2 ∫ a b r 2 d θ ) s m a l l l o o p ] Substitute ( 1 2 + cos θ ) for r in the above equation. A = [ ( ∫ 0 2 π 3 ( 1 2 + cos θ ) 2 d θ ) − ( ∫ 2 π 3 π ( 1 2 + cos θ ) 2 d θ ) ] = [ ( ∫ 0 2 π 3 ( ( 1 2 ) 2 + ( cos θ ) 2 + 2 ( 1 2 ) cos θ ) d θ ) − ( ∫ 2 π 3 π ( ( 1 2 ) 2 + ( cos θ ) 2 + 2 ( 1 2 ) cos θ ) ) d θ ] = [ ( ∫ 0 2 π 3 ( 1 4 + cos 2 θ + cos θ ) d θ ) − ( ∫ 2 π 3 π ( 1 4 + cos 2 θ + cos θ ) ) d θ ] = 1 4 [ ( ∫ 0 2 π 3 ( 1 + 4 cos 2 θ + 4 cos θ ) d θ ) − ( ∫ 2 π 3 π ( 1 + 4 cos 2 θ + 4 cos θ ) ) d θ ] Substitute 1 + cos 2 θ 2 for cos 2 θ in the above equation

Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)

2nd Edition

ISBN: 9781285101552

Author: James Stewart

Publisher: Cenage

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Chapter 9.4, Problem 27E

To determine

To find: The area inside the larger loop and outside the smaller loop of the limacon r=12+cosθ.

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Chapter 9.4, Problem 26E

Chapter 9.4, Problem 29E

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Chapter 9 Solutions

Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)

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 area inside the larger loop and outside the smaller loop of the limacon r = 1 2 + cos θ .

Solution Summary: The author calculates the area inside the larger loop and outside the smaller loop of the limacon using an online graphing calculator.

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To find: The area inside the larger loop and outside the smaller loop of the limacon r=12+cosθ.

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Chapter 9 Solutions