Double Integrals in Polar Coordinates

Transcription

1 Double Integrals in Polar Coorinates Part : The Area Di erential in Polar Coorinates We can also aly the change of variable formula to the olar coorinate transformation x = r cos () ; y = r sin () However, ue to the imortance of olar coorinates, we erive its change of variable formula more rigorously. To begin with, the Jacobian eterminant (x; (r; ) = cos () r sin () sin () r cos () = r cos () + r sin () = r As a result, the area i erential for olar coorinates is A (x; (r; ) r = rr Let us consier now the olar region S e ne by = ; = ; r = g () ; r = f () where f () an g () are containe in [; q] for all in [; ] : If ; : : : ; m is an h- ne artition of [; ] an r ; : : : ; r n is an h- ne artition of [; q] ; then the image of [; ] [; q] is a artition of the image of the region with near arallelograms whose areas are enote by A jk : Since the area i erential is A = rr, the area of the near arallelogram is aroximately A jk r j r j j so that if x jk = r j cos ( k ) an y jk = r j sin ( k ) ; then lim nx h! j= k= mx (x jk ; y jk ) A jk = lim nx h! j= k= mx (r j cos ( k ) ; r j sin ( k )) r j r j j

2 Writing each of these limits as ouble integrals results in the formula for change of variable in olar coorinates: Z Z f() (x; y) A = (r cos () ; r sin ()) rr () g() To ai in the use of (), let us notice that if is constant, then r = is a circle of raius centere at the origin in the xy-lane, while if is constant, then = is a ray at angle beginning at the origin of the xy-lane. Moreover, the origin corresons to r = : EXAMPLE Use () to evaluate Z Z x x + y yx Solution: To o so, we transform the iterate integral into a ouble integral Z Z x x + y yx = x + y A where is a sector of a circle with raius : In olar coorinates, is the region between r = an r = for in [=4; =]: Since r = x + y ; the ouble integral thus becomes x + y A = Z = Z r rr = Z = Z =4 =4 r 3 r

3 an the resulting iterate integral is then easily evaluate: x + y A = Z = =4 r 4 4 = Z = =4 = 4 Check your eaing: What oes y = x correson to in olar coorinates? Areas an Volumes in Polar Coorinates If is a region in the xy-lane boune by = ; = ; r = g () ; r = f () ; then () imlies that Area of = A = Z Z f() thus allowing us to n areas in olar coorinates. g() rr EXAMPLE Fin the area of the region between x =, x = ; y =, an y = x Solution: Since x = corresons to r cos () = or r = sec () ; the region is between the line r = sec () an a circle of raius from = to = =4: Thus, the area of the region is Area = A = Z =4 Z sec() rr 3

4 an evaluation of the iterate integral leas to Area = = Z =4 r Z =4 sec() sec () = ( tan ())j=4 = 4 Moreover, we can use olar coorinates to n areas of regions enclose by grahs of olar functions. EXAMPLE 3 What is the area of the region enclose by the carioi r = + cos () ; in [; ] : Solution: Since the carioi contains the origin, the lower bounary is r = : Thus, its area is Area = Z Z +cos() rr = Substituting an exaning leas to Area = = = = 3 Z Z Z r +cos() + cos () + cos () + cos () + + cos () 3 + sin () + sin () 4 4

5 Polar coorinates can also be use to comute volumes. equation of a shere of raius centere at the origin is x + y + z = For examle, the Solving for z then yiels shows us that the shere can be consiere the soli between the grahs of the two functions g (x; y) = x y ; f (x; y) = x y over the circle x + y = in the xy-lane. Since circle x + y = e nes the tye I region x = y = x x = y = x the volume of the shere of raius is given by the iterate integral V = Z Z x x x y yx () EXAMPLE 4 Use olar coorinates to evaluate V = Z Z x x x y yx Solution: To begin with, we rewrite the iterate integral as a ouble integral over the interior of the circle of raius centere at the origin, which is often enote by D: Z Z V = (x + y ) A D In olar coorinates, the isc D of raius is boune by the curves = ; = ; r = ; r = ; so that Z Z V = x y A = D Z Z r rr 5

6 Thus, if we let u = r ; then u = rr; u () = ; u () = ; so that V = = = Z Z Z Z u = u u 3= 3= 3= 3 = Check your eaing: What is the volume of the unit shere? Ineenent Normal Distributions In statistics, a normally istribute ranom variable with mean an stanar eviation has a Gaussian ensity, which is function of the form (x) = (x ) e =( ) (3) It follows that the joint ensity for two ineenent, normally istribute events is a function of two variables of the form (x; y) = (x) (y) = e (x ) =( ) e (x ) =( ) For simlicity, we will consier here only ineenent, normally istribute events with mean = in both an stanar eviations = = : In such cases, the joint ensity function is (x; y) = e (x +y )=( ) EXAMPLE 5 Let (X; Y ) be the coorinates of the nal resting lace of a ball which is release from a osition on the z-axis towar the xy-lane, an suose the two coorinates are ineenently normally istribute with a mean of an a stanar eviation of 6

7 3 feet. What is the robability that the ball s nal resting lace will be no more than 5 feet from the origin? Solution: Since = 3; the joint ensity function is (x; y) = 8 e (x +y )=8 an we want to know the robability that (X; Y ) will be in a circle with raius 5 centere at the origin. Since such a circle corresons to r = to r = 5 for in [; ] ; the robability is P X + Y 5 = Converting to olar coorinates then yiels P X + Y 5 = 8 8 e (x +y )=8 A Z Z 5 e r =8 rr an if we now let u = r ; u = rr; then u () = an u (5) = 5 imlies that P X + Y 5 = = Z Z 5 Z = e 5=8 = :75648 e u=8 u 8 8e 5=8 7

8 Thus, there is about a 75% chance that the ball s nal resting lace will be no more than 5 feet from the origin. Check your eaing: How exactly o we interret P X + Y 5? An Imortant esult in Statistics Finally, the value of the integral I = Z e x x is very imortant in statistical alications. To evaluate it, we rst notice that Z Z Z Z I = e x x e y y = e x e y yx That is, I is a tye I iterate integral which can be converte to olar coorinates. EXAMPLE 6 Evaluate the integral I = Z Z e x e y yx Solution: To o so, let us notice that Z Z I = e (x +y ) A Qua I However, in olar coorinates, the rst quarant is given by r = to r = for = to = =: Thus, I = As a result, we can write I = Z = Z = Z " Z lim! e r rr e r rr # 8

9 Thus, if we let u = r ; u = rr; u () = ; u () = ; then " # I = = = = 4 Z = Z = Z = h lim! lim! Thus, I = =; which imlies both Z e e u u e i Z e x x = an Z e x x = Exercises: Evaluate the following iterate integrals by transforming to olar coorinates.. x x + y yx. x tan y x yx 3. x x yx 4. x y x +y yx x +y 5. y y xy x +y 6. y x xy x +y 7. 4 x 4 x 9 x y yx 8. 4 x x y x +y yx 9. x x x yx. x y x +y x yx x +y. x x x +y yx. x x yx x +y 3. y x x +y xy 4. y y x +y xy 5. x x yx [x +y ] 3= 6. y xy [x +y ] 3= Each of the following olar curves encloses a region that contains the origin. 9

10 Fin the area of the region the curve encloses. 7. r = 5; in [; ] 8. r = 3; in [ ; ] 9. r = sin () ; in [; ]. r = 4 cos () ; in [; ]. r = ; in [; ]. r = jj + ; in [ ; ] 3. r = sin (3) ; in [; =3] 4. r = 4 cos (3) ; in [; =3] 5. r = sin (5) ; in [; =5] 6. r = sin () ; in [; ] 7. r = + cos () ; in [; ] 8. r = + sin (3) ; in [; =3] 9. r = sin () + cos () ; in [; ] 3. r = 3 sin () + 4 cos () ; in [; ] 3. Use olar coorinates to n the volume of a right circular cone with height h an a circular base with raius (hint: the equation of the cone is z = h x + y 3. A right circular cone with a base of raius is slice by a lane of the form z = h + (h h ) x + where h an h are ositive. What is the shae of the soli between this lane an the xy-lane, an what is its volume? 33. ecall that if < " < an > ; then r = " cos () is an ellise which encloses a region. Evaluate A [x + y ] 3= 34. Evaluate the ouble integral ya where is the olar ellise escribe in exercise In examle 4, what is the robability that

11 . (a) The nal osition of the ball is in the st quarant an is no more than 5 feet from the origin. (b) The nal osition of the ball is between 3 an 7 feet from the origin. (c) The nal osition of the ball is in the xy-lane. 36. After several throws at a art boar, a art thrower ns that both the X an Y coorinates of his arts have a mean of an a stanar eviation of 3 inches. What is the robability that a ranomly selecte art throw from all those he has thrown will be in the "bulls eye", if the bulls eye is a circle of raius one inch centere at the origin? 37. Suose an airlane has two rocket engines whose time of ignition with resect to a time zero is normally istribute with a stanar eviation of = : secons. If the rockets ignitions are ineenent events, what is the robability that the sum of the squares of the ring times is less than.? 38. In exercise 37, what is the robability that the left engine will re no more than 3 times later than the right engine? 39. The antennae lengths of a samle of 3 woolice were measure an foun to have a mean of 4 mm an stanar eviation of.37 mm. Assuming the antennae lengths are normally istribute, what is the robability of one of the antennae of a woolice being twice as long as the other? (Hint: substitute to translate the means to ). 4. Acme sheet metal rouces several hunre rectangular sheets of metal each ay. If errors in the lengths an withs of the rectangular sheets are ineenent ranom variables with mean of an a stanar eviation of s =. inches, then what is the robability that the error in the area of the rectangular sheets excees. inches? 4. Use the metho in the iscussion receing examle 6 to evaluate J = Z x e x x 4. Fin the area an the centroi of a carioi of the form r = + cos () 43. Write to Learn: A freezer rouces ice cubes with normally istribute temeratures with a mean of F an a stanar eviation of F: Write a short essay in which you comute an exlain the robability that two ice cubes chosen at ranom will have temeratures that i er by no more than 3 F; assuming the temeratures are ineenent. 44. Try it out! Dro a ball several times (i.e., -3 times) from a osition irectly above an origin in an xy-lane you create. (Hint: to avoi any bias, you might want to secure the ball with a threa an then release the ball by cutting the threa). Suose that (x ; y ) ; (x ; y ) ; : : : ; (x n ; y n )

12 enotes the nal stoing oints of the ball. The samle means of both the x s an the y s shoul be ractically zero. The samle stanar eviation for the x s is s Pn j= x = (x j x) n an the samle stanar eviation y for the y s is similar. Show that x y an then reeat examle 5 using the samle stanar eviations as the value for :