tapia jat4858 HW07 clark 52990 This print out should have 24 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 region is given by Z o 1 2 n I 2 cos 2 cos 2 d 2 2 3 2 10 0 points The shaded region in 1 Z 2 cos2 d 2 To evaluate this last integral we use the double angle formula cos2 1 1 cos 2 2 For then Z Z 2 1 2 2 1 cos 2 d cos d 2 2 2 i 2 1h 1 1 sin 2 2 2 2 2 Consequently the shaded region has area lies inside the polar curve r 2 cos and outside the polar curve r cos Determine the area of this region 1 area 3 1 2 2 area 3 3 1 4 2 3 1 4 area 2 2 3 area 3 5 area correct 4 6 area 3 2 3 4 Are there any other ways of calculating this area without using integration keywords area polar cooordinates definite integral circle 002 10 0 points Find the area of the region bounded by the polar curve r 2 ln 3 as well as the rays 1 and e 1 3e 1 correct 2 1 2 area 3e 2 2 1 area 3 area 3e 1 Explanation As the graphs show the polar curves intersect when 2 cos cos 4 area i e at 2 Thus the area of the shaded 6 area 1 3e 2 4 5 area 3e 2 1 3e 1 4 tapia jat4858 HW07 clark 52990 Explanation The area of the region bounded by the graph of the polar function r f as well as the rays 0 1 is given by the integral Z 1 1 A f 2 d 2 0 5 area 4 Explanation The area of the region bounded by the graph of the polar function r f and the rays 0 1 is given by the integral When f 2 ln 3 1 A 2 0 1 1 e therefore the area of the enclosed region is thus given by the integral Z 1 e A 2 ln 3 2 d 2 1 Z 1 e 2 ln 3 d 2 1 2 Z 1 f 2 d 0 On the other hand the graph of r 2 cos is the cardioid similar to the one shown in To evaluate this last integral we use Integration by Parts for then ie 1 Z e 1h 2 ln 3 2 d A 2 1 2 1 ie 1h 2 ln 1 1 2 Consequently area A 003 1 3e 1 2 so in this case we can take 0 0 and 1 2 Thus the area of the region enclosed by the graph is given by the integral 1 A 2 10 0 points 2 Z 0 2 cos 2 d Now Find the area of the region enclosed by the graph of the polar function 2 cos 2 4 4 cos cos2 r 2 cos 1 area 7 2 since 2 area 19 2 But then 3 area 3 4 area 9 correct 2 1 9 4 cos cos 2 2 2 cos2 2 1 1 cos 2 2 1 9 4 cos cos 2 d 2 2 0 i2 1 1h 9 4 sin sin 2 0 2 2 4 1 A 2 Z tapia jat4858 HW07 clark 52990 Consequently area A The area of the region bounded by the graph of the polar function r f and the rays 1 and 2 is given by the integral Z 1 2 f 2 d A 2 1 9 2 When keywords polar graph area cardioid 004 3 f cos 1 1 2 2 10 0 points therefore 1 A 2 Find the area of the shaded region inside the polar curve Z cos2 d 4 On the other hand r cos cos2 shown in 1 1 cos 2 2 Thus y 1 A 4 x 2 1 Z when 1 4 and 2 3 1 area 1 1 3 48 8 2 area 1 7 2 3 48 16 3 area 7 1 2 3 correct 48 16 4 area 1 1 2 3 48 16 5 area 1 7 3 48 8 6 area 1 1 2 3 48 16 3 1 cos 2 d 4 i 3 1 1h sin 2 4 2 4 Consequently area Explanation 3 7 1 2 3 48 16 keywords polar graph polar integral double angle 005 10 0 points Find the area of the shaded region tapia jat4858 HW07 clark 52990 inside the graph of since 2 area 3 area 4 area 5 area 1 1 cos 2 2 cos2 r 1 2 cos 1 area 4 Hence 3 3 correct 4 1 3 3 2 4 3 3 4 2 3 2 1 A 2 Z 2 3 0 3 4 cos 2 cos 2 d i2 3 1h 3 4 sin sin 2 2 0 Consequently 3 3 area A 4 6 area 1 3 3 7 area 2 4 Explanation The area of a region bounded by the graph of the polar function r f and the rays 0 1 is given by the integral Z 1 1 f 2 d A 2 0 keywords polar graph area cardioid polar integral 006 10 0 points Find the area of the shaded region in In this question the function is r 1 2 cos on the other hand to determine the rays 0 and 1 bounding the shaded region note that as ranges from 0 to 2 3 the graph i begins on the x axis when 0 at r 3 ii crosses the y axis when 2 at r 1 iii first passes though the origin when r 0 i e when 2 3 Thus the area of the shaded region is given by the integral Z 1 2 3 A 1 2 cos 2 d 2 0 specified by the graphs of the circles r 2 cos 1 area 2 4 1 2 area 1 But 1 2 cos 2 1 4 cos 4 cos2 3 4 cos 2 cos 2 3 area 2 4 area 1 2 r 2 sin tapia jat4858 HW07 clark 52990 5 Consequently the shaded region has 5 area 1 correct 2 area 6 area Explanation The area of a region bounded by the graph of polar function r f between the rays 0 1 is given by the integral Z 1 1 A f 2 d 2 0 Now the graph of r 2 cos is a circle centered on the x axis while that of r 2 sin is a circle centered on the y axis in addition both pass through the origin and have the same radius So the circles intersect also at 4 as the figure shows 1 2 keywords definite integral area between curves polar area circle 007 10 0 points Which one of the following integrals gives the arc length of the portion shown as solid blue in the graph The area of the shaded …