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All Textbook Solutions for Calculus (MindTap Course List)
25E26ERectangular-to Polar Conversion In Exercises 2534, convert the rectangular equation to polar form and sketch its graph. x2+y2=a228E29E30E31E32E33ERectangular-to-Polar Conversion In Exercises 25-34, convert the rectangular equation to polar form and sketch its graph. (x2+y2)29(x2y2)=035E36E37E38E39E40E41E42E43EPolar-to-Rectangular Conversion In Exercises 35-44, convert the polar equation to rectangular form and sketch its graph. r=cotcsc45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103ERotated Curve In Exercises 103-105, use the results of Exercises 101 and 102. Write an equation for the rose curve r=2sin2 after it has been rotated counterclockwise by an angle of (a) =/6,(b) =/2, (c) =2/3, and (d) =. Use a graphing utility to graph each rotated rose curve.105EProof Prove that the tangent of the angle (0/2) between the radial line and the tangent line at the point (r, ) on the graph of r=f() (see figure) is given by tan=|rdr/d| q107E108E109E110E111E112E113E114E115E116EArea of a Polar Region What should you check before applying Theorem 10.13 to find the area of the region bounded by the graph of r=f()?Points of Intersection Explain why finding points of intersection of polar graphs may require further analysis beyond solving two equations simultaneously.Area of a Polar Region In Exercises 3-6, write an integral that represents the area of the shaded region of the figure. Do not evaluate the integral. r=4sinArea of a Polar Region In Exercises 3-6, write an integral that represents the area of the shaded region of the figure. Do not evaluate the integral. r=cos25E6EFinding the Area of a Polar Region In Exercises 7-18, Find the area of the region. Interiorofr=6sin8EFinding the Area of a Polar Region In Exercises 7-18, Find the area of the region. One petal of r=2cos310E11EFinding the Area of a Polar Region In Exercises 7-18, Find the area of the region. Three petals of r=cos513E14E15E16E17E18E19E20E21E22E23E24E25E26E27EFinding Points of Intersection In Exercises 27-34, find the points of intersection of the graphs of the equations. r=3(1+sin)r=3(1sin)29E30E31E32EFinding Points of Intersection In Exercises 27-34, find the points of intersection of the graphs of the equations. r=2r=234E35E36E37E38E39EFinding the Area of a Polar Region Between Two CurvesIn Exercises 37-44, use a graphing utility to graph the polar equations. Find the area of the given region analytically. Common interior of r=53sin and r=53cos41E42E43E44E45E46E47EFinding the Area of a Polar Region Between Two CurvesIn Exercises 45-48, find the area of the region. Common interior of r=acos and r=asin, where a049E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74ESurface Area of a Torus Find the surface area of the tours generated by revolving the circle given by r=2 about the line r=5sec.Surface Area of a Tours Find the surface area of the tours generated by revolving the circle given by r=a about the line r=bsec, where 0ab.77E78E79ELogarithmic Spiral The curve represented by the equation r=aeb, where a and b are constants, is called a logarithmic spiral. The figure shows the graph of r=e/6,22. Find the area of the shaded region.81E82E83E84E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54EArea of a Region In Exercises 55-58, use the integration capabilities of a graphing utility to approximate the area of theregion bounded by the graph of the polar equation. r=32cos56E57E58E59E60E61E62E63E64E65EComet Hale-Bopp The comet Hale-Bopp has an elliptical orbit with the sun at one focus and has an eccentricity of e0.995. The length of the major axis of the orbit is approximate 500 astronomical units. (a) Find the length of its minor axis. (b) Find a polar equation for the orbit. (c) Find the perihelion and aphelion distances.Eccentricity In Exercises 67 and 68, let r0 represent the distance from a focus to the nearest vertex, and let r1 represent the distance from the focus to the farthest vertex. Show that the eccentricity of an ellipse can be written as e=r1t0r1+r0 Then show that r1r0=1+e1e.68EWriting Vectors in Different Forms In Exercises 1 and 2, let u=PQ and v=PR and (a) write u and v in component form, (b) write u and v as the linear combination of the standard unit vectors i and j, (c) find the magnitudes of u and v, and (d) find 3u+v. P=(1,2),Q=(4,1),R=(5,4)2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32REFinding a Unit VectorFind a unit vector that is orthogonal to both u=2,10,8 and v=4,6,8.34RE35REVolumeUse the triple scalar product to find the volume of the parallelepiped having adjacent edges u=2i+j, v=2j+k, and w=j+2k.37RE38RE39RE40RE41RE42RE43RE44RE45RE46REDistance Find the distance between the planes 5x3y+z=2 and 5x3y+z=3.48RE49RE50RE51RE52RE53RE54RE55RE56RE