COMPLETE Chapter 15 Multiple Integrals. Section 15.1 Double Integrals Over Rectangles. Section 15.2 Iterated Integrals

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1 Mat 7 Calculus III Updated on /3/7 Dr. Firoz COMPLT Chapter 5 Multiple Integrals Section 5. Double Integrals Over ectangles amples:. valuate the iterated integral a) (5 ) da, {(, ), } and b) (4 ) da, [,] [,] Solution: a) b) (5 ) da (5 ) dd [5 / ] () 4.5 (4 ) da (4 ) dd [4 ] 3. If [,] [,], show that sin( + ) da Solution: On, + < π,sin( + ). Thus we have sin( + ) da da Section 5. Iterated Integrals Fubini s Theorem If f (, ) is differentiable on {(, ) a b, c d} b d then f (, ) da f (, ) dd f (, ) dd d b a c c a amples valuate the integral in two different was using Fubini s theorem (-4) dd d ( + ) [ + / 3] [ / 3+ / 3 ] 44

2 Mat 7 Calculus III Updated on /3/7 Dr. Firoz also dd d ( + ) [ / 3 + ] [ + ] 44 e dd + dd ( ) + + dd 5. valuate cos( + ) da, {(, ) π, π / } + 6. valuate, {(, ), } da + 7. Find the volume of the solid in the first octant bounded b the clinder z 9 and the plane Solution: On the plane z 9 3 in the first octant. Thus we have 3 (9 ) 36 V dd 8. Find the average value of (, ) f e + e, [,4] [,]. The average value is given b fave f (, ) da A( ), where A() represents the area of the region. Now 4 fave e e dd + (Tr to solve the integral) valuate (5 ) dd. Ans: From problem 9. draw the bounded region b z 5, {(, ) 5, 3} and evaluate 3 5 (5 ) dd geometricall valuate (4 ) dd. Ans:. valuate da over the rectangle {(, ) 3, }. Ans: -5/6 3. valuate cos( ) cos ( π ) da; [,/ ] [, π ] Ans: /(3 π ).

3 Mat 7 Calculus III Updated on /3/7 Dr. Firoz 4. valuate Section 5.3 Double Integral Over General egions amples. valuate v. 3. dd Answer: 3/ v dudv Answer: /3 da over the region enclosed between /,,, 4 4. ( ) da, D {(, ) +, +, 3} D Answer: /6 Answer: -68/3 5. Use double integral to find the volume of the tetrahedron bounded b the coordinate planes and the plane given b z 4 4 V (4 4 ) dd Answer: 4/3 6. Find the volume of the solid bounded b the clinder + 4 and the planes + z 4, z Answer: 6 π 7. valuate the following multiple integrals: a) V ( + ) dd Answer: 4/35 b) ( + 3 / 3 V ) dd Answer: 4/35 8. valuate + ( ) da, where is bounded b the curves, 3 and compare our result with eample 7. Answer: 4/35 9. Sketch the region of integration and change the order of integration of 4 / f (, ) dd Answer: f (, ) dd. Homework # : + V 3 ( ) dd 9 /. Homework # 3: V ( + ) dd 3dd 6 / / 4. Homework # 38: V f (, ) dd 4 f (, ) dd 4

4 Mat 7 Calculus III Updated on /3/7 Dr. Firoz 3. Homework # 56: 3 + 4) da da A( D) (5) smmetr. Section 5.4 Polar Coordinates If f is continuous on a polar rectangle given b then f (, ) da β α π amples D ( using β b α a D f ( r,θ ) rdrdθ a r b, α θ β where. valuate sin θ da, where is the region in the first quadrant that is outside the circle r and inside the cardioid r ( + cosθ ) sinθ da π / (+ cosθ ) sinθrdrdθ 8/ 3. The sphere of radius a centered at origin is epressed in rectangular coordinates as + + z a and hence its equation in clindrical coordinates is r the sphere. + z a. Use this equation and a polar double integral to find the volume of π a, θ 4 / 3πa r + z a z ± a r V a r rdrd 3. Use polar coordinate to evaluate draw a diagram and set 4. Homework Problems: π / 3 4) f (, ) da ( + ) 3 / dd 3/ 4 ( + ) dd r drdθ f ( r cosθ, r sinθ ) rdrdθ 7) One loop of the rose cos3θ π π / 6 cos 3θ r : A da 8) Area of the region enclosed b r 4 + 3cosθ, D π / 6 π / 5 rdrdθ π / π 4+ 3cosθ D {( r, θ ) θ π, r 4 + 3cosθ}, A da rdrdθ 4/ π D 3

5 Mat 7 Calculus III Updated on /3/7 Dr. Firoz Section 5.7 Triple Integrals amples. valuate z dv where is defined as a region bounded b, 3, z. Draw he bounded region and set d d z dv z dz 648. valuate zdv where is the wedge in the first octant that is cut from the clindrical solid + z b the planes, Draw diagram and set z dv zdzdd 3 / 8 using tpe I region 3. valuate zdv where is the solid tetrahedron bounded b the four planes,, z, + + z V zdv zdzdd / 4 4. valuate dv where is the solid tetrahedron bounded b the four planes,, z, + + z 4 Answer: V 4/3 5. Use triple integral to find volume of the tetrahedron bounded b the four planes,, z, + + z Answer: V /3 6. Find the volume of the tetrahedron with vertices (,, ), (,, ), (,, ) and (,, ). Use Tpe I, Tpe II and Tpe III regions. Find the plane through (,, ), (,, ), and (,, ) which is Tpe I: V Tpe II: V Tpe III: V / ( f (, dzdd / z / f (, dddz z+ 7. Homework Problems: f (, dddz / ( z. f (, dzdd, from solid f (, ddzd, from z - solid z+ f (, ddzd, from z - solid

6 Mat 7 Calculus III Updated on /3/7 Dr. Firoz ) z zdv zddzd /, use the plane z thru (,, ), (,, ) and (,, ) 9 6. z dv z dzdd /3 / 9 7 / f (, dv /3 / / / / / f dzdd f dzdd / 4 / 3 z 4 / 4 /3 z 4 / z /3 z 4 / /3 4 z z f ddzd f dddz /3 9 / z 9 /3 9 / z 9 /3 z / z 9 /3 / 9 z z f ddzd f dddz 34. Tpe I, solid {(,,, z } {(,,, z } Tpe II, z solid {(,, z } {(, z, z, } Tpe III, z solid {(,, z, z } {(, z, z, z } 48. f ave f z dv V (,, ) ( ) polar coordinate. Now f ave V ( ) f (, dv π, where ( ) V dzdd π / using ( + ) dzdd / Section 5.8 Triple Integrals in Clindrical and Spherical Coordinates Clindrical Coordinates:

7 Mat 7 Calculus III Updated on /3/7 Dr. Firoz β h ( θ ) u ( r, θ ) f (, dv α h ( θ ) u ( r, θ ) Spherical Coordinates: f ( r, θ, rdzdrdθ d β b f (, dv amples c α a f ( ρ, θ, φ) ρ sinφdρdθdφ. Use triple integral in clindrical coordinates to find the volume of the solid that is bounded b the hemisphere and laterall b the clinder z 5 below bounded b the plane π 3 5 r z dv dzdd rdzdrdθ π / valuate b clindrical coordinate: dzdd dzdd 3 9 π 3 9 r 3 r dzdrd 3. Use spherical coordinate to evaluate 4 4 z z dzdd valuate cos θ θ 43 π / z z dzdd + + π π / 5 ρ φ φd ρdφdθ π cos sin 64 / ( + ) dv where is the solid in the first octant that lies beneath the paraboloid z r θ z θ π r z r {(,, ) /,, } and π / r 3 4 ( + ) dv r cosθdzdrdθ / Find the volume of the solid that lies within both the clinder sphere + + z 4 r θ z θ π r r z r {(,, ),, 4 4 } and π 4 r V rdzdrdθ 4 π / 3(8 3 3) 4 r +, and the

8 Mat 7 Calculus III Updated on /3/7 Dr. Firoz 6. Use spherical coordinates to evaluate ( + ) dv where H is the H hemispherical region that lies above the plane and below the sphere + + z H π π / valuate ( + ) dv ρ sin φdρdθdφ 4 π /5 ( + + z ) + + z e dddz Use spherical coordinate and find 8. ( + + z ) π π 3 ρ + + z e dddz lim ρ e sinφd ρdθdφ π Section 5.9 Change of Variables in Multiple Integrals amples. Given / 4( u + v), / ( u v) and T is the transformation from uv plane to plane. Find a) T (,3) b) Sketch the constant v curve corresponding to v,,,, c) Sketch the constant u curve corresponding to u,,,, d) Sketch the image of the square region in uv plane under the transformation T to the plane. a) T ( u, v) (, ) u, v 3,, thus T (,3) (, ) b) Solving for u and v we have u + v For given v,,,, we find,,,,. You can plot all these equations in the plane. c) We have u +. For given u,,,, we find +, +, +, +, +. You can plot all these equations in the plane. d) Tr ourself.

9 Mat 7 Calculus III Updated on /3/7 Dr. Firoz. valuate da, where is the region bounded b +,, +, + 3. Direct evaluation is complicated, we use substitution like u + v u,3, v, and / ( u + v), / ( u v) The Jacobian Now / / J ( u, v) / / / 3 u da J ( u, v) dudv / 4ln 3 + v 3. valuate e da, where is the region bounded b /,, /, / Write u /, v, v e da e dvdu e e u / / ( )ln 4. Find the volume of the region enclosed b the ellipsoid + + z a b c Use the transformation (substitution) u / a, v / b, w z / c u + v + w, which is a sphere of radius. Now dv J ( u, v, w) dudvdw 4 π abc / 3. For volume of a sphere see eample, S section Find the image of the set S which is a disk given b u + v under the transformation au, bv. 6. valuate ( 3 ) da, where is the triangular region with vertices (, ), (, ) and (, ). The line thru (, ) and (, ) is /, which is the image of v The line thru (, ) and (, ) is, which is the image of u and the line thru (, ) and (, ) is + 3, which is the image of u + v The Jacobian Now J ( u, v) 3 u ( 3 ) da ( u 5 v)3dvdu 3

10 Mat 7 Calculus III Updated on /3/7 Dr. Firoz 7. valuate ( + ) da, where is the region bounded b the ellipse + ; under the substitution u / 3 v, u + / 3v Solution: + u + v, The Jacobian J ( u, v) 4 / 3 u ( + ) da ( u + v )4 / 3 dudv 4 π / 3 u 8. valuate cos da, where is the trapezoidal region with vertices (, ), + (, ), (, ), (, ). Use u, v +, J ( u, v) /, cos da 3/ sin + 9. valuate sin(9 + 4 ) da, where is the region bounded b the ellipse Use u 3, v J ( u, v) / 6 π / sin(9 + 4 ) da / 6 sin( u + v ) dudv / 6 r sinr drdθ S