Mat 267 Engineering Calculus III Updated on 03/21/11. with length xi xi x. max x 0
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1 Mat 67 ngineering Calculus III Updated on // Chapter Multiple Integrals Section. ouble Integrals Over ectangles ecall from calculus I (Mat 65 that the definite integral f ( d gives the area under the curve f ( on [a, b] and is estimated b the iemann sum b a * i n i f * ( i where we take n subintervals [ i, i] with length i i i and is a sample point in these subintervals. Then we have the integral b a n * ( lim ( i ma i f d f (if this limit eists. Volume and double integrals: In the similar manner we consider a function of two variables z on a closed rectangle defined b [ a, b] [ c, d] {(, a b, c d } And we first assume that. The double integral m n * * (, lim ( ij, ij ij ma (, i j f da f A finds the volume of the solid under the surface z and on the domain (if this limit eists. i i Area under the curve ( Volume under the surface (
2 Mat 67 ngineering Calculus III Updated on // If {, j,r j }, j =,,m, is an h- fine tagged partition of [ a,b], and if {, k,t k }, k =,,n is a l- fine tagged partition of [ c,d], then the rectangles [ j, j ] [ k, k ] partition the rectangle [a,b] [ c,d] and the points (r j,t k are inside the rectangles [ j, j ] [ k, k ]. The figure above shows how to make partitions on the given domain. 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 ]. If [,] [,], show that sin( da Solution: On,,sin(. Thus we have sin( da da Iterated Integrals Fubini s Theorem If is differentiable on {(, a b, c d } b d then da dd dd d b a c c a amples valuate the integral in two different was using Fubini s theorem (-4.. also dd d 4 4 dd d ( [ / ] [ / / ] 44 ( [ / ] [ ] 44
3 Mat 67 ngineering Calculus III Updated on //.. 4. e dd ( dd dd 5. valuate cos( da, {(,, / } 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 in the first octant. Thus we have (9 6 V dd 8. Find the average value of e e, [,4] [,]. The average value is given b fave A ( da, where A( represents the area of the region. Now 9. valuate 4 fave e e dd (Tr to solve the integral 5 (5 dd. Ans: 7.5. From problem 9. draw the bounded region b z 5, {(, 5, } and evaluate 5. valuate. valuate. valuate (5 dd geometricall. 4 (4 dd. Ans: da over the rectangle {(,, }. Ans: -5/6 cos( cos ( da; [,/ ] [, ] Ans: /(.
4 Mat 67 ngineering Calculus III Updated on // Section. ouble Integral Over General egions amples. valuate.. 4a. v dd Answer: / v dudv Answer: / da over the region enclosed between /,,, 4 Answer: /6 ( da, {(,,,, } 4b. ( da, {(,,, } Answer: Not possible, no bounded domain. Answer: -68/ 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, plot the domain Answer: 4/ 6. Find the volume of the solid bounded b the clinder 4 and the planes z 4, z Answer: 6 7. valuate the following multiple integrals: a V ( dd Answer: 4/5 b V ( dd Answer: 4/5 8. valuate / ( da, where is bounded b the curves, and compare our result with eample 7. Answer: 4/5 9. Sketch the region of integration and change the order of integration of / 4 dd Answer:. Find the volume of the solid under the surface bounded b and. Solution hint: V 4 dd z and above the region ( dd 9/ 4
5 Mat 67 ngineering Calculus III Updated on //. Find the volume of the solid enclosed b the parabolic clinder and the planes z z. Solution hint: V ( dd dd 6/ 5. Sketch the region of integration and change the order of integration of 4 4 dd. Solution hint: 4 4 /4 4 dd dd. Use smmetr to evaluate ( 4 da, where is the region bounded b the square with vertices ( 5, and (, 5 Solution hint: Section. Polar Coordinates ( 4 da da A( (5 using smmetr. If f is continuous on a polar rectangle given b a r b, where then amples. valuate sin da f ( r, rdrd b a da, where is the region in the first quadrant that is outside the circle r and inside the cardioid r ( cos sin da / ( cos sin rdrd 8/. 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 z a. Use this equation and a polar double integral to find the volume of the sphere. r z a z a r a, V a r rdrd 4/ a. Use polar coordinate to evaluate draw a diagram and set ( ( / / dd dd r 4 drd / 5 5
6 Mat 67 ngineering Calculus III Updated on // 4. Find the area of the region given: a between the circles of radius and in the first quadrant. Solution: / da f ( r cos, r sin rdrd b Use double integral to find the area of one loop of the rose r cos : Solution hint: A da / 6 / 6 cos rdrd / c Use double integral to find the area of the region enclosed b r 4 cos, {( r,, r 4 cos } 4 cos Solution hint: A da rdrd Section.5 Triple Integrals 4/ amples. valuate z dv where is defined as a region bounded b,, z. raw he bounded region and set. valuate zdv z dv d d z dz 4 where is the wedge in the first octant that is cut from the clindrical solid z b the planes, raw diagram and set. valuate zdv V,, z, z zdv 4. valuate dv zdv zdzdd / 8 using tpe I region where is the solid tetrahedron bounded b the four planes zdzdd / 4 where is the solid tetrahedron bounded b the four planes,, z, z 4 Answer: V = 4/ 5. Use triple integral to find volume of the tetrahedron bounded b the four planes,, z, z Answer: V = / 6
7 Mat 67 ngineering Calculus III Updated on // 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 / ( / dzdd z dddz z / dddz / ( z. dzdd from solid ddzd from z - solid z ddzd, from z - solid 7. valuate the triple integrals: a zdv. Where is the solid tetrahedron with vertices (,,, ((,,, (,, and (,,. Solution hint: raw the diagram and find the plane b z thru (,,, (,, and (,, z zdv zddzd / use zdv. Where is the solid bounded b the clinder planes,, z in the first octant. Solution hint: z dv 9 z dzdd 7/ 8 z 9 and the c press the integral dv in si different was, where is the solid bounded b the surface given b Solution hint: dv / / z 9 4 / / f dzdd = / / = / / / 4 / z 4 / 4 / z 4 f dzdd f ddzd = / z / z 4 / / 4 z z f dddz 7
8 Mat 67 ngineering Calculus III Updated on // = = / 9 / z 9 / 9 / z 9 / z / z 9 / / 9 z z f ddzd f dddz d Write five other integrals that are equal to dzdd Solution hint: Tpe I, solid {(,,, z } {(,,, z Tpe II, z solid {(,, z } {(, z, z, Tpe III, z solid {(,, z, z } {(, z, z, z 48. Find the average value of the paraboloid Solution hint: V ( f ave V ( z z over the region enclosed b z and the plane f V ( ave dzdd dv z. z dv, where / using polar coordinate. Now ( dzdd / } } } Section.6-.7 Triple Integrals in Clindrical and Spherical Coordinates Clindrical Coordinates: dv Spherical Coordinates: h ( u ( r, h ( u ( r, f ( r,, rdzdrd dv d b f (,, sin d d d amples c a. 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 9 8
9 Mat 67 ngineering Calculus III Updated on // r z dv dzdd rdzdrd 9 /. valuate b clindrical coordinate: 9 9 dzdd 9 9 r. Use spherical coordinate to evaluate valuate 4 z z dzdd the paraboloid ( 9 9 dzdd 9 r cos dzdrd 4 / z z dzdd / cos sin d d d 64 / 9 5 dv where is the solid in the first octant that lies beneath z {( r,, /, r, z r } and / r 4 ( dv r cos dzdrd / 5 5. Find the volume of the solid that lies within both the clinder sphere z 4 {( r,,, r, 4 r z 4 r } and 4 r V rdzdrd 4 r 4 / (8 6. Use spherical coordinates to evaluate H ( dv where H is the hemispherical region that lies above the plane and below the sphere z H / 4 ( dv sin d d d 4 /5, and the 7. valuate ( z z e dddz Use spherical coordinate and find ( z z e dddz lim e sin d d d 9
10 Mat 67 ngineering Calculus III Updated on // Section.8 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 (, 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,, thus T (, (, 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.. valuate da, where is the region bounded b,,,. irect evaluation is complicated, we use substitution like u v u,, v, and / ( u v, / ( u v The Jacobian Now. valuate u v / / J ( u, v / / / u v u da J ( u, v dudv / 4ln v e da, where is the region bounded b /,, /, /
11 Mat 67 ngineering Calculus III Updated on // Write u /, v, v e da e dvdu e e u / / ( ln z 4. Find the volume of the region enclosed b the ellipsoid 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 abc /. For volume of a sphere see eample, S section. page # Find the image of the set S which is a disk given b transformation au, bv. u v under the 6. valuate ( 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, which is the image of u v The Jacobian Now 7. valuate u v J ( u, v u v u ( da ( u 5 vdvdu ( da, where is the region bounded b the ellipse ; under the substitution /, / Solution: u v u v u v u v, The Jacobian J ( u, v 4 / u v u ( da ( u v 4 / dudv 4 / u 8. valuate cos da, where is the trapezoidal region with vertices (,, (,, (,, (,. Use u, v, J( u, v /, cos da / sin
12 Mat 67 ngineering Calculus III Updated on // 9. valuate sin(9 4 da, where is the region bounded b the ellipse 9 4. Use u, v J( u, v / 6 / sin(9 4 da / 6 sin( u v dudv / 6 r sin r drd S
MULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then
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