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# Class Note for MATH 1432 at UH

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Area of a Polar Region 39 Area of a Polar Region 0 polar axis O K polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 2 14 Area of a Polar Region 39 E quot Area of a Polar Region The area of the polar region F generated by rp6 ag g a A 5992 0 9 0 polar axis 0 g polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 2 14 Area of a Polar Region l W quot Area of a Polar Region The area of the polar region F generated by rp6 ag g 31 A Z glp9l2d9 0 polar axis 0 g polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 2 14 Area of a Polar Region l W quot Area of a Polar Region The area of the polar region F generated by rp6 ag g 31 A Z glp9l2d9 Let P 6061 o Hn be a partition of 04 Set r agigig p6 and R arggag 06 is 0 polar axis 0 g polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 2 14 Area of a Polar Region quot Area of a Polar Region V The area of the polar region F generated by rp6 ag g 31 A Z glp9l2d9 Let P 6061 o Hn be a partition of 04 Set r agelg 06 and R argea p6 Then is 0 polar axis 1 1 ErI2A6 g A g EREAH 0 g polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 2 14 Area of a Polar Region quot Area of a Polar Region V The area of the polar region F generated by rp6 ag g is 31 A Z glp9l2d9 Let P 6061 o Hn be a partition of 04 Set r agelg 06 and R argea p6 Then 0 polar axis 1 1 ErI2A6 g A g EREAH Summing from i 1 to i n yields 1 LAP A s Ufa with me We 0 g polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 2 14 Area of a Polar Region quot Area of a Polar Region V The area of the polar region F generated by rp6 agdg 31 A Z glp9l2d9 Let P 6061 o Hn be a partition of 04 Set r agelg 06 and R argea p6 Then is 0 polar axis 1 1 ErI2A6 g A g EREAH Summing from i 1 to i n yields 1 LAP A s Ufa with me We Since P is arbitrary we conclude a a A f6d6 p62d6 0 g polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 2 14 Area of a Polar Region 39 Basic Polar Area Circles Ribbons Flowers Limacons 2 Area of a Circle of Radius a A 7ra r22 r 2 r4sin0 1 2 40036 Circle in Polar Coordinates ra O 0 2w Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region 39 Basic Polar Area Circles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra 2 e r 2 r4sin0 1 2 40036 Circle in Polar Coordinates r a O S 0 S 27r 5 A 2L p02 d0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region 39 Basic Polar Area Circles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra 2 e r 2 r4sin0 1 2 40036 Circle in Polar Coordinates r a O S 0 S 27r 5 27139 Azfa p02d00 a2d0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region 39 Basic Polar Area Circles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra 2 e r 2 r4sin0 1 2 40036 Circle in Polar Coordinates r a O S 0 S 27r B 27139 A2 lp02de lla2d9la2 2 a2 oz 2 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region 39 Basic Polar Area Circles Ribbons Flowers Limacons 2 Area of a Circle of Radius a A 7T3 r22 r 2 r4sin0 1 2 40036 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 Area of a Polar Region 39 Basic Polar Area Circles Ribbons Flowers Limacons 2 Area of a Circle of Radius a A 7ra m r 2 r4sin0 1 2 40036 Circle in Polar Coordinates r a O S 0 S 27r 5 A A p02 d0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Basic Polar Area Circles Ribbons Flowers Limacons 2 Area of a Polar Region Area of a Circle of Radius a A 7ra m EV r 2 r4sin0 1 2 40036 Circle in Polar Coordinates r a 0lt10ltlt2w Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 314 Basic Polar Area Circles Ribbons Flowers Limacons 2 Area of a Polar Region Area of a Circle of Radius a A 7ra m EV r 2 r4sin0 1 2 40036 Circle in Polar Coordinates r a fg0fg2w 5 1 27 1 1 A p02d0 a2d0 a227r7ra2 a 2 0 2 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 314 Area of a Polar Region l 39 Basic Polar Area Cimles Ribbons Flowers Limacons 2 Area of a Circle of Radius a A 7ra r22 r 2 r4sin6 1392 40030 Circle in Polar Coordinates r2asin0 0 0 7T Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 Area of a Polar Region l 39 Basic Polar Area Cimle Ribbons Flowers Limacons Area of a Circle of Radius a A 7T3 r 2 r4sin6 1392 40030 r2asin0 0 0 7T Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region l 39 Basic Polar Area Cimles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra r22 r 2 r4sin6 1392 40030 Circle in Polar Coordinates r2asin0 0 0 7r B 7139 Azfa p02d00 2asin02d0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region l 39 Basic Polar Area Cimles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra r22 r 2 r4sin6 1392 40030 Circle in Polar Coordinates r2asin0 0 0 7r 5 W 7139 A2 lp02de l2asin02de2a2sin20de a 2 0 2 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region l 39 Basic Polar Area Cimle Ribbons Flowers Limacons Area of a Circle of Radius a A 7T3 r22 r 2 r4sin6 1392 40030 r2asin0 0 0 7T Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region l 39 Basic Polar Area Cimle Ribbons Flowers Limacons Area of a Circle of Radius a A 7T3 r22 r 2 r4sin6 1392 40030 r2asin0 0 0 7T Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Basic Polar Area Cimlea Ribbons Flowers Limacons A7Ta2 Area of a Polar Region l adius a Area of a Circle of R r22 r 2 r4sin6 1392 40036 Circle in Polar Coordinates W 3 r 2acos E S 0 S 7 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston February 25 2008 314 Area of a Polar Region l 39 Basic Polar Area Cimles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra mm r 2 r4sin6 1392 40036 Circle in Polar Coordinates r 2acos ES 03 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region l 39 Basic Polar Area Cimles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra Dl r 2 r4sin6 1392 40036 Circle in Polar Coordinates r 2acos ES 03 37 5 A2 p02d0f2 l 23COSQ2d9 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Area of a Polar Region l 39 Basic Polar Area Cimles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra Dl r 2 r4sin6 1392 40036 Circle in Polar Coordinates 7T 37T 2 lt lt r acosQ 20 2 g1 3 71 37 2 A2 p02d0 2ac0502d02a2 c0520d6 oz E Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 25 2008 3 14 Basic Polar Area Circles Ribbons Flowers Limacons Area of a Polar Region I Area of a Circle of Radius a A 7ra Del r 2 r4sin6 1392 40036 Circle in Polar Coordinates W 3 r 2acos S 0 S 2 2 37F 37F Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston February 2amp 2008 314 Area of a Polar Region i 39 Basic Polar Area Cimles Ribbons Flowers Limacons Area of a Circle of Radius a A 7ra Dl r 2 r4sin6 1392 40036 Circle in Polar Coordinates 51 311 3 Az Elp0l2d0f2El 2ac050l2d022a22c0520d6 a 9 9 2 377i 1 1 2 377i 1 2 7T 2 2a l cos20 d022a d02a 7ra 2 2 g 2 2 H Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 2amp 2008 3 14 Area of a Polar Region r i I Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24cos 26 Sketch r2 32 cos 20 in 4 stages 0 E g w Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 3926 29018 4 14 Area of a Polar Region r i I Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24cos 26 Sketch r2 32 cos 20 in 4 stages 0 E 1 w 274 5 A p02d0 6 Ill Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 3926 29018 4 14 Area of a Polar Region r i I Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24cos 26 Sketch r2 32 cos 20 in 4 stages 0 g a371 w T 4gt21 5 A p02d040 a2cos26d0 oz Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 3926 29018 4 14 Area of a Polar Region I 7 Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24cos 26 Sketch r2 32 cos 20 in 4 stages 0 E g w 4gt21 B1 2 1 2 2 1 2 A p0 d04 a cos20d02a sm 20 a 2 0 2 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 3926 2MB 4 14 Area of a Polar Region r i I Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24COS 26 Ribbon Sketch r2 a2 sin 20 in 4 stages 0 E g w Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 3926 29018 4 14 Area of a Polar Region r i I Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24COS 26 Ribbon Sketch r2 a2 sin 20 in 4 stages 0 E g w A awe 1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 3926 29018 4 14 Area of a Polar Region r i I Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24COS 26 Ribbon Sketch r2 a2 sin 20 in 4 stages 0 E g w 3 A4ap02d040 a2sin26d6 I Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 3926 29018 4 14 Area of a Polar Region r r i A Basic Polar Area Circles Ribbons Flowers Limacons Area of a Lemniscat Ribbon A a2 r24sin 26 r24COS 26 Ribbon Sketch r2 a2 sin 20 in 4 stages 0 g g w 181 2 2 2 1 E 2 A24 p0 d024 a sin26d6 2a cos20 a 04 2 0 2 2 0 F1 Jiwen He University of Houston Math 1432 Section 25526 Lecture 13 February 3926 29018 4 14 1725mi 52 mm mm z emmsaie 2 zzssmum Wm WM zemme Area of a Polar Region Area of a Flower A rsin 36 r cos 46 Sketch r sin 36 in 6 stages 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 614 Area of a Polar Region 2 939 Area of a Flower 1 3 39 3 87r 1 7z39 6 7r 1 1 rsin 36 rcos 46 Sketch r sin 36 in 6 stages 0 57T B A imam Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 6 14 Area of a Polar Region w 139 Area of a Flower A l 3 7Z39 3 87r 1 7z39 6 7T 1 rsin 36 rzcos40 Sketch r sin 36 in 6 stages 0 57T A Lp62d660 sin362d6 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 6 14 Area of a Polar Region A Area of a Flower r cos 46 rsin 36 Sketch r sin 36 in 6 stages 0 57T A Lp62d660 sin362d6 11 1 236 cos66 0 2 2 Math 1432 Section 26626 Lecture 13 d6 11 614 February 26 2008 Jiwen He University of Houston Area of a Polar Region p5 iquot 139 Area of a Flower l 3 7Z39 3 87r 1 7r 6 7r 1 1 rsin 36 rzcos46 Sketch r sin 36 in 6 stages 0 27T 5W 577T 7 A Lp62d660 sin362d6 11 1 3 1 g 6 cos66d6 6 sin66 3 H 0 2 2 2 6 O 4 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 6 14 Area of a Polar Region 2 939 Area of a Flower 1 3 7Z39 3 8n 1 7z39 6 7T 1 1 rsin 36 rzcos 46 Flower Sketch r cos46 in 16 stages 0 a 15 27r A 16 p62d6 oz Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 6 14 Area of a Polar Region y Area of a Flower A r cos 46 rsin36 Sketch r cos46 in 16 stages 0 a 15 27r 1 A 16 p62d6 168 cos462d6 O 111 614 February 26 2008 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston Area of a Polar Region 1 3 7Z39 3 8n 1 7z39 6 7T 1 1 rsin 36 rzcos 46 Flower i Sketch r cos46 in 16 stages 0 a 15 27r 1 A 16 p62d6 168 cos462d6 O Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 6 14 Area of a Polar Region A Area of a Flower r cos 46 r sin 36 Flower Sketch r cos46 in 16 stages 0 a 157 27r 1 8 cos 46 20 6 1 8 1 6 sin86 3 8 0 2 1 1 cos86d6 4 0 2 2 February 26 2008 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston 614 Area of a Polar Region Limagons Snails r c058 a 2 1 3 1 r30036 r cose r10050 r cose convex limagon cardioid limagon with limagon with a dimple an inner loop Sketch r a c058 a Z 1 in 2 stages 07T 7t27t Jiwen He University of Houston Math 1432 Section 25625 Lecture 13 February 26 2008 77 14 Area of a Polar Region Limagons Snails r c058 a 2 1 3 1 r30036 r cose r10050 r cose convex limagon cardioid limagon with limagon with a dimple an inner loop Sketch r a c058 a Z 1 in 2 stages 07T 7t27t Jiwen He University of Houston Math 1432 Section 25625 Lecture 13 February 26 2008 77 14 Area of a Polar Region Limagons Snails r c058 a 2 1 3 1 r30036 r cose r10050 r cose convex limagon cardioid limagon with limagon with a dimple an inner loop Sketch r a c058 a Z 1 in 2 stages 07T 7t27t 711 2 A 0 ac056 d6 Jiwen He University of Houston Math 1432 Section 25625 Lecture 13 February 26 2008 77 14 Area of a Polar Region Limagons Snails r c058 a 2 1 r30036 rcos 9 r10050 rcose convex limagon cardioid limagon with limagon with a dimple an inner loop Sketch r a c058 a Z 1 in 2 stages 07T 7t27t A ac0562d6322acos6cos26d6 0 0 Jiwen He University of Houston Math 1432 Section 25625 Lecture 13 February 26 2008 77 14 Area of a Polar Region Limagons Snails r c058 a 2 1 3 1 r30036 r cose r10050 r cose convex limagon cardioid limagon with limagon with a dimple an inner loop Sketch r a c058 a Z 1 in 2 stages 07T 7t27t A ac0562d6322acos6cos26d6 0 0 a22ac056cos26d6 I 0 Jiwen He University of Houston Math 1432 Section 25625 Lecture 13 February 26 2008 77 14 Area of a Polar Region Limagons Snails r c058 a 2 1 3 1 r30036 r cose r10050 r cose convex limagon cardioid limagon with limagon with a dimple an inner loop Sketch r a c058 a Z 1 in 2 stages 07T 7t27t A ac0562d6322acos6cos26d6 0 0 7T 7T2 l l 2 l 2 l 0a 2acos 22cos26d6 a 260 a 27T I Jiwen He University of Houston Math 1432 Section 25625 Lecture 13 February 26 2008 77 14 Area of a Polar Region Limaeon Snail 1 2 co 0 1 O Q I polaraXIs 37r V z 1 2 c056 1 OSQSEZT OSQS 39 J Q pobraxS Dwaraxs 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t Area Within Outer Loop Acuter February 256 2003 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston Area of a Polar Region Limaeon Snail 1 2 co 0 1 O Q I polaraXIs 37r V z 1 2 c056 1 OSQSEZT OSQS 39 J Q polar axis 0 polar axis 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t Ainner 0 A Aouter Area Within Outer Loop Acuter Aouter February 256 2003 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston Area of a Polar Region Limaeon Snail 1 2 co 0 1 O quot polaraxis 37r V z 1 2 c056 1 OSQSEZT OSQS 39 J a polar axis 0 polar axis 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t Ainner 0 A Aouter Area Within Outer Loop Acuter Acuter 2 gll 2cos 92d6 3 February 256 2003 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston h miampiu liih LmLMa r1 2co Area of a Polar Region ELM polar axis Limaeon Snail 0 1 O 39 quot39 polar aXIs OSQSE 2 2 Olt6ltl 37T b J polar axis polar axis OE 3 5 0933 Ogeszn Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7T7T 7t 37T27t o A Aouter Ainner Area Within Outer Loop Acuter 7T1 7T Acuter 2 E 1 2cos l2d6 f 1 4cosi9 4cos2 i9d6 E 00 February 256 2003 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston F10 polar axis r 1 2 co Area of a Polar Region 37r Limaeon Snail Q r 1 2 c056 b 1 O S 6337239 J polar axis 0 polar axis 0 9327 5 03933 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7r 37r27t Ainner o A Aouter Area Within Outer Loop Acuter 1 2cosr92d6 f 1 4cosi9 4cos2 i9d6 Aouter 21 2 3 1 4cos 922cos26 d6 00 7T February 256 2003 Math 1432 Section 26626 Lecture 13 25 Jiwen He University of Houston 00gtl Area of a Polar Region r 1 2 co Limaeon Snail 0 1 O quot polar axis 3 7r z 1 2 c056 1 OSQSEZT OSQS 39 J Q polar axis 0 polar axis 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t o A Aouter Ainner Area Within Outer Loop Acuter 1 4COS9 4cos2 6 d6 7T1 Aouter E 2C062d6 If 1 4COS9 2 2cos26 d6 36 4sin6 sin 26 H a a 1 Math 1432 Section 26626 Lecture 13 February 26 2008 8 14 Jiwen He University of Houston Area of a Polar Region Limaeon Snail r 1 2co m V 37r 39 polar axis 0 1 O 39 quot39 polar aXIs z 1 2 c056 Oses n Osesn J Q polar axis polar axis 0 3 O 3 HS 27 5 03933 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t 27l l Ainner 0 A Aouter Ainner Area Within Outer Loop Acuter Aouter 2COS92d9 a 1 4cos 922cos26 d6 7T 1 4COS9 l 4COS2 6 d6 7T 36 4sin 6 sin 26 February 256 2003 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston 814 Area of a Polar Region Limaeon Snail 1 2 co 0 1 O Q I polaraXIs 37r V z 1 2 c056 1 OSQSEZT OSQS 39 J Q pobraxS Dwaraxs 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t 0 A Aouter Ainner 27T i Ainner Area Within Inner Loop Ainner 39 814 February 256 2003 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston Area of a Polar Region Limaeon Snail 1 2 co 0 1 O Q I polaraXIs 37r V z 1 2 c056 1 OSQSEZT OSQS 39 J Q pobraxS Dwaraxs 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t 0 A Aouter Ainner 27T i Ainner Area Within Inner Loop Ainner 39 Ainner February 26 2008 I 8 l4 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston Area of a Polar Region Limaeon Snail 1 2 co 0 1 O I quot39 polaraXIs 37r V z 1 2 c056 1 OSQSEZT OSQS 39 J Q polar axis 0 polar axis 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t 0 A Aouter Ainner 27l l Ainner Area Within Inner Loop Ainner 39 Ainner 2 11 2cos 92d6 0 oogt1 February 26 2008 I 8 14 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston meiwiu ewliww LmLMa r1 2co Area of a Polar Region H 0 V Limaeon Snail my Q z 1 2 c056 b 1 OSQSEZT p polar axis polar axis OE 3 5 03633 039327 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t 0 A Aouter Ainner 27l l Ainner Area Within Inner Loop Ainner r 1 2 g 2 1 2cos 9 d6 1 4cos94cos 6 d6 0 814 Ainner O polar axis oogt1 February 256 2003 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston Area of a Polar Region r 1 2 co Limaeon Snail 0 1 O quot polar axis 3 7r z 1 2 c056 1 OSQSEZT OSQS 39 J Q pobraxS Dwaraxs 0952 0997 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7r7r 7t 37r27t 0 A Aouter Ainner 27T i Ainner Area Within Inner Loop Ainner 39 gl 2 g 2 Ainner2 1 2cos 9 d6 1 4cos94cos 6 d6 0 0 814 February 256 2003 31 4cos 922cos26 d6 0 Math 1432 Section 26626 Lecture 13 Jiwen He University of Houston Area of a Polar Region i Limacon Snail r 1 2co 3 polar axis 3 If polar axis 1 OSQSEZT OSQS 39 polar axis polar axis 5 3 Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7T7T 7t 37T27t 0 A Aouter Ainner 27l l Ainner Area Within Inner Loop Ainner 39 1 Ainner 2g 1 2cos l2d6 31 4cos 94cos2 6 d6 0 0 1 31 4cos 22cos26 d6 36 4sin6 sin26 H 0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 I 8 14 Area of a Polar Region i Limagon Snail r 1 2co Area between the Inner and Outer Loops 0 Sketch in 4 stages 0 7T7T 7t 37T27t 0 A Aouter Ainner 27T i 7T 7739 i Area Within Inner Loop Ainner 39 1 Ainner 2g 1 2cos 2d6 31 4cos 4cos2 6 d6 0 0 1 31 4cos922cos26 d6 36 4sin6 sin26 in 0 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 I 8 14 Area between Polar Curves Between Polar Curves Between Circles CircleampFlower Circleamp Area between Polar Curves Qz Area between r 2 010 and r 2 020 area of Q polar axis Jiwen He University of Houston Math 1432 r Section 26525 Lecture February 26L 28 9 14 Area between Polar Curves Between Polar Curves Between Circles CircleampFlower Circleamp Area between Polar Curves Qz Area between r 2 010 and r 2 020 52 1 2 area on 020 d0 2 2 51 50192 d9 polar axis Jiwen He University of Houston Math 1182 a Section 26625 Letture February 25L 28 9 14 Area between Polar Curves Between Polar Curves Between Circles CircleampFlower Circleamp Area between Polar Curves Qz Area between r 2 010 and r 2 020 52 1 2 area on 020 d0 2 2 51 a 5mm d0 1 5 ip29i2 p102 d9 if 061O 2067 315213 polar axis Jiwen He University of Houston Math 132 a Se tio n 26625 Letture February 26 28 9 14 Area between Polar Curves etween Polar CuNes Between Circles CircleampFlower Circleamp Area between Polar Curves Qz Area between r 2 010 and r 2 020 52 1 2 area of Q 5 p20 d0 a2 51 a 50192 d9 1 5 lp29l2 p10l2 d9 if 05120522057 315213 polar axis Extra care is needed to determine the intervals of 0 values eg 051 31 and 052 g over which the outer and inner boundaries of the region are traced out I Jiwen He University of Houston Math 1432 a Section 25526 LeviWe February 26 quot 28 9 a 14 Area between Polar Curves Area between Circles r 2c Area between r 2cost9 and r 1 2 c056 polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 1014 Area between Polar Curves Area between Circles r 2c Area between r 2cost9 and r 1 o The two intersection points 6 2C061 gt cos gt 5 7r gt 0473 l7 r2c056 polar axis Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 10 14 Area between Polar Curves Area between Circles r 2c Area between r 2cost9 and r 1 o The two intersection points 2cos 1 gt c056l 15 gt Ob li7 o By symmetry r 2 056 area of 9 gt polar axis Jiwen He University of Houston Math 1432 Saction 26626 Lecture 13 February 26 10 14 r2c056 Area between r 2cos6 and r 1 o The two intersection points 2cos6 1 gt cos6 gt 15 gt Og 7 o By symmetry area of Q 1 polar axis Jiwen He University of Houston ip29i2 iP19i2gt d9 2 Math 13432 Section 26626 Lecture 1339 February 26 10 14 r 2 c056 Area between r 2cos6 and r 1 o The two intersection points 2cos6 1 gt cos6 gt 15 gt Og 7 o By symmetry area of Q 1 Jiwen He University of Houston polar axis we we de 20lt2cos62 1 d6 2 Math 13432 Section 26626 Lecture 1339 February 26 10 14 r2c056 Area between r 2cos6 and r 1 o The two intersection points 2cos6 1 gt cos6 gt 15 gt Og 7 o By symmetry area of Q 1 Jiwen He University of Houston polar axis we we de 20lt2cos62 1 d6 7T 3 f 1 2cos26 d6 0 2 Math 13432 Section 26626 Lecture 1339 February 26 10 14 r 2 c056 Area between r 2cos6 and r 1 o The two intersection points 2cos6 1 gt cos6 gt 5 397r gt Og 7 o By symmetry area of Q 1 Jiwen He University of Houston polar axis we we de 20lt2cos62 1 d6 7T 3 f 1 2cos26 d6 0 2 E 3 6sin 26 g 7T 3 0 Math 13432 Section 26626 Lecture 1339 February 26 10 14 3 monzzma s 2 b z 2 a madam 272 5 z Area between Polar Curves r25in6 r25in 26 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 12 14 r25in6 r25in 26 Jiwen He University of Houston 0 The three intersection points 2sin26 2sin6 gt 2sin6 cos sin6 gt sin6 2cos 9 1 O gt sin6 O or c056 gt O or 5 gt 053 O and 053 57r Math 1432 Section 26626 Lecture 13 February 26 2008 1214 i Area between Polar Curves Area outsideCircle r 2 sin 6 and inside flower r 2 si 26 o The three intersection points 2sin26 2sin6 gt 2sin6 cos sin6 gt sin6 2cos 9 1 O gt sin6 O or cos gt O or 5 gt 053 O r25in6 and 0573 577T B A o By symmetry A1 r25in 26 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 12 14 i Area between Polar Curves Area outsideCircle r 2 sin 6 and inside flower r 2 si 26 o The three intersection points 2sin26 2sin6 gt 2sin6 cos sin6 gt sin6 2cos 9 1 O gt sin6 O or cos gt O or 5 gt 053 O and 053 57r B A o By symmetry A1 3 WW WW d6 r25in6 r25in 26 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 12 14 i Area between Polar Curves Area outsideCircle r 2 sin 6 and inside flower r 2 sin 26 o The three intersection points 2sin26 2sin6 gt 2sin6 cos sin6 gt sin6 2cos 9 1 O gt sin6 O or cos gt O or 5 gt 053 O r25in6 and 0573 577T B A o By symmetry A1 3 WW WW d6 r2sm2e f 2sin26 2 2sin6 2 d6 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 12 14 439 Area between Polar Curves Area outsideCircle r 2 sin 6 and inside flower r 2 sin 26 o The three intersection points 2sin26 2sin6 gt 2sin6 cos sin6 gt sin6 2cos 9 1 O gt sin6 O or c056 gt O or 5 gt 053 O r25in6 and 0573 577T A o By symmetry K A1 ff p29i2 919i2 d 9 g 1 2 2 r2in29 0 E 2sm26 2sm6 1 3 C cos 26 cos 46 d8 0 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 12 14 Area outside Ci o The three intersection points 2sin26 2sin6 gt 2sin6 cos sin6 gt sin6 2cos 9 1 O gt sin6 O or cosmic mew and 063 57T r25in6 o By symmetry d9 r2in26 If E 25in O 2 C cos 26 cos 46 d8 0 lsin26 1sin46 g 2 4 o 8 oogt1 oogt1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008 12 14 Area between Polar Curves Between Polar Curves Between Circles CircleampFlower Circlegt Area between Circle r 25in 6 and Limagon r g sin 6 2 Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 28 2008 13 l4 Area between Polar Curves Area between Circle r 25in 6 and Limagon r sin 0 The two intersection points 2sin6 sin6 gt sin6 gt Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 2 2008 13 l4 Area between Polar Curves Area between Circle r 25in 6 and Limagon r 5 sin o The two intersection points 05 39 39 39 l 1 57T 2 2snn6 2 snn6gtsnn6 2gt6 6 o The area can be represented as follows A Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 2 2008 13 l4 o The two intersection points 2sin6 sin6gtsin6gt 577 W o The area can be represented as follows 7 Ag 2sin62d6 O Jiwen He University of Houston Math 1432 w Section 26626 Lecture 13 February 2 2008 13 14 o The two intersection points 2sin6 sin6gtsin6gt 577 W o The area can be represented as follows 7 Ag l2sine2a6 0 2 7T 1 S sin62d6 6 DII I Jiwen He University of Houston Math 1432 w Section 26626 Lecture 13 February 2 2008 o The two intersection points 39 39 39 l 57T 2snn6 2 snn6gtsnn6 2gt6 6 o The area can be represented as follows 7 Azfgl2sin62d6 0 2 57139 13 2 Sln6 5 2sin62d6 6 cnll 1 Jiwen He University of Houston Math 1432 w Section 26626 Lecture 13 February 2 2008 13 14 o The two intersection points 39 39 39 l 57T 2snn6 2 snn6gtsnn6 2gt6 6 o The area can be represented as follows 7 Azfgl2sin62d6 0 2 57139 13 2 Sln6 w 2sin62d6 6 cnll 5 15 2 3 47T 8 Jiwen He University of Houston Math 1432 w Section 26626 Lecture 13 February 2 2008 13 14 a Am 039 My R290quot 4 am Mme des mm a was MW e Am balmquot my CurIs BeMeen My Cums BeMeen des BeMeen 0mg 2nd F wev BeMeen Ordeznd Wm Lecture 13 Section 95 Area in Polar Coordinates Jiwen He Department of Mathematics University of Houston jiwenhe mathuhedu httpmathuhedumjiwenheMath1432 0 g polar axis m Jiwen He University of Houston Math 1432 Section 26626 Lecture 13 February 26 2008

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