is a surface above the xy-plane over R.

Transcription

1 Chapter 13 Multiple Integration Section 13.1Double Integrals over ectangular egions ecall the Definite Integral from Chapter 5 b a n * lim i f x dx f x x n i 1 b If f x 0 then f xdx is the area under the curve (and above the x-axis) between x a and x b. Volume and Double Integral a Suppose z f x, y defined on x, y a x b, c y d. If f x, y 0, then z f x, y is a surface above the xy-plane over. Let S be the solid that lies above and under z. S x, y, z 0 z f x, y, x, y Subdivide the region into rectangles, x, length and y,width, then area of each rectangle, ij, * * * * is A x y now the approximate height of the solid above ij is f xij, y ij where xij, y ij is a * * point in ij V f x, y A. and ij ij n m So *, * n m V f xij yij A and V lim f x *, * ij yij A. i1 j1 nm, i 1 j 1 Definition: The double integral of f over is If the limit exists. n m, lim *, * ij ij f x y da f x y A nm, i 1 j 1 If f x, y 0 then the volume of the solid S is V, f x y da. ecall from Chapter 6, to find the volume of a solid of revolution we made slices perpendicular to the x-axis, xb xa V A x dx Where Axis the cross-sectional area at x. Now the area of the cross-section is: yd, A x f x y dy yc 1

2 To evaluate hold x fixed and integrate with respect to y (partial integration with respect to y). b b d b d V A x dx f x y dy dx f x y dydx a a c a c So,, Example: Find the volume of the solid under f x, y x y and over the rectangle x, y x 4, 1 y 1. Theorem 13.1: (Fubini) Double Integrals on ectangular egions Let f x, y be continuous on the rectangular region : a x b, c y d. The double integral of f over may be evaluated by either of two iterated integrals: Example: Calculate, d b b d,,, f x y da f x y dxdy f x y dydx. c a a c f x y da for f x, y x y and x, y 0 x 1,1 y Example: Evaluate x 1 xy da. Where x, y 0 x 4,1 y

3 f x y da will be positive if f x, y is above the region on the xy-plane; it is negative if Note:, f x, y is below the region on the xy-plane; it can also be zero. Definition: Average Value of a Function Over a Plane egion The average value of an integrable function f over a region is 1 f f x, yda area of Example: Find the average value of the function f x, y sin xsin y over the region x, y 0 x,0 y. Section 13. Double Integrals over General egions Suppose f x, y is defined over the bounded region D (i.e. D can be enclosed in a rectangular region. Let F x, y, if, f x y x y D then 0 if x, y but not D 3

4 D,, f x y da F x y da Any will do as long as it contains D. If f x, y 0 then the V f x, yda D Note F may have discontinuities at the boundary of D. Nonetheless, if f is continuous on D and the boundary curve is well behaved then F x, yda exist and therefore f x, yda exists. D Type I: Suppose D x, y a x b, g x y g x, then f x, yda, Example: Evaluate, y x and y x. D 1 b gx f x y dydx. D a g x f x y da where f x, y x y where D is the region bounded by x 0, 1 Type II: Suppose D x, y c y d, h y x h y then f x, yda, 1 D d h y c h y 1 f x y dxdy Example: Find the volume of the solid bounded above by f x, y x y over the region bounded by y x, x 1 in the first quadrant. 4

5 Example: Evaluate, y x and yx. f x y da where f x, y y x where D is the region bounded by y, D 4x y xe Example: For dydx, sketch the region of integration, reverse the order of integration and y evaluate the integral. Example: Find the volume of the tetrahedron bounded by the coordinate planes and the plane z 8 x 4y. 5

6 Example: Find the volume of the wedge sliced from the cylinder x y 1 by the planes z1 x and zx 1. Area by Double Integral Definition: The area of a closed bounded plane region is Example: Find the area of the region bounded by the curves x y 1 and xy. A da Example: y dxdy gives the area of a region in the xy-plane. Sketch the region, label each 1 y bounding curve with its equation, and give the coordinates of the points where the curves intersect. Find the area. 6

7 Section 13.3 Double Integrals in Polar Coordinates ecall Polar Coordinates from Chapter 10 r x y, x rcos, y rsin Polar ectangle r, a r b, ecall the area of sector of a circle is 1 A r What is the area of the polar rectangle? m n m n * * * *, lim, lim i cos j, i sin j f x y da f x y da f r r da n, m n, m j1 i1 j1 i1 m n * * * i j lim g r, r r g r, rdrd nm, j 1 i 1 a b b f r cos, rsin rdrd, where a Area in Polar Coordinates The area of a closed bounded region in the polar coordinate plane is A rdrd 7

8 1 1 y Example: Evaluate 0 0 x y dxdy Example: Evaluate y1 xy dxdy Example: Find the average height of the (single) cone z x y above the disk x y a in the xy-plane. 8

9 Example: Find the area of the region that lies inside the cardioid circle r 1 r 1 cos and outside the Section 13.4 Triple Integrals in ectangular Coordinates Triple Integral T,, lim,, f x y z dv f x y z V P 0 T is a solid body with density function f. Limit exists as norm P (diagonal of block) approaches zero provided f is continuous on T and that the boundary region of T is reasonably well behaved. Example: Evaluate,, 0 x, 0 y 3, 0 z 1. f x y z dv for f x, y, z x y z and T is a rectangular box where T 9

10 Suppose T is described by z x, y z z x, y, g x y g x, a x b then 1 1 b g x z x, y,,,, f x y z dv T a g x z x, y 1 1 f x y z dzdydx Example: Evaluate for f x, y, z x y and T is the region bounded between the surfaces z x z x and 0 y 3. Definition: The volume of a closed, bounded region D in space is V dv Example: Find the volume bounded by the surfaces z x, yz 4, y 0 and z 0 D Example: Find the volume bounded by the planes z 0, x 0, y, and z y x. Note: dv dzdydx dzdxdy dydzdx dydxdz dxdzdy dxdydz 10

11 Definition: The average value of a function F over a region D in space is 1 favg f x, y, zdv VD D Where VD is the volume of the region D. Section 13.5 Triple Integrals in Cylindrical and Spherical Coordinates Definition: Cylindrical coordinates represent a point P in space by ordered triples r,, z in which 1. r and are polar coordinates for the vertical projection of P on the xy-plane.. z is the rectangular vertical coordinate. Equations elating ectangular x, y, z and Cylindrical r,, z Coordinates x rcos, y rsin, z z r x y, tan y / x 11

12 To integrate over T x, y, z x, yd, h x, y z h x, y where D is given in polar coordinated by g h r cos, rsin 1,, D r g r g Then,, cos, sin, T g h r cos, rsin f x y z dv f r r z rdzdrd Example: Evaluate cylinders x y 4 and x y 9 xdv, where T is enclosed by the planes z 0 and z x y 3 and by the T Spherical Coordinates A point P in space is,, 1. is the distance from P to the origin.. is the angle OP makes with the positive z-axis is the angle from cylindrical coordinates. Equations elating Spherical Coordinates to Cartesian and Cylindrical Coordinates r sin, x rcos sin cos z cos, y rsin sin sin x y z r z Note: c is a sphere centered at the origin with radius c. c is a half plane c is either the top half of a cone if 0 / and the bottom half if / 1

13 Example: Plot the point, / 3, / 4 then find the rectangular coordinates. Example: Change the point with rectangular coordinates 1,1, 6 to spherical coordinates. Example: Change the point with cylindrical coordinates 6, / 4, to spherical coordinates. Example: Identify the surface sin. Example: Identify the surface cos. Example: Write the equation x y z z 0 in spherical coordinates. 13

14 Example: Write the equation x y x in spherical coordinates. To integrate f x, y, zdv in spherical coordinates: T dv sind dd and T,,,, g, g, So T min max 1 max g,,, sin cos, sin sin, cos sin min g1, f x y z dv f ddd Example: Evaluate x y z e dv for the sphere x y z T 9 in the first octant. 14

15 Section 13.6 Integrals for Mass Calculations To determine the mass and center of mass for a thin plate (lamina) with variable density (mass/unit area) at point xy, in of xy,. Mass: M, x y da First Moments:,,, y x M x x y da M y x y da The first moment measures the tendency of the object to rotate about the line (axis). M y M x Center of Mass: x, y M M Example: Find the mass and center of mass for the region bounded by y x, y 0, and x 1 with density x, y x. The center of mass will be on a line of symmetry. However, when the density function is not constant, BE CAEFUL!. The region and the density function must both be symmetric to the line of symmetry. For example: Consider a right triangle with legs of length L on the positive x-axis and y-axis. This region is symmetric about the line y x. a. If xy, 1 will the center of mass be on the line? b. If r, r will the center of mass be on the line? c. If x, y y will the center of mass be on the line? 15

16 The mass and center of mass for an object occupying a region T in space with density x, y, z : Mass: M,, T x y z dv Moment: M x x, y, z dv, M y x, y, z dv, M z x, y, z dv, yz xz xy T T T Myz M M xz Center of Mass: x, y, z M M M xy Example: A solid trough of constant density is bounded below by the surface z 4y, above by the plane z 4, and on the ends by the planes x 1 and x 1. Find the center of mass. Section 13.7 Change of Variables in Multiple Integrals ecall from Chapter 5: If x g u then the single integral xb xa ugb uga ' f x dx f g u g u du Suppose we want to transform the double integral f x, yda to make it easier to evaluate. ecall polar form: x rcos and y rsin f x y da f r r rdrd where S is, then, cos, sin the region in the r -plane corresponding to the region in the xy-plane Suppose G is region in the uv-plane is transformed one-to-one into the region in the xy-plane by the equations of the form x gu, v and y hu, v If g, h, and f have continuous partial derivatives and J u, v is zero only at isolated points, if at all, then,,,,, G f x y da f g u v h u v J u v dudv S 16

17 Where J u, v x x u v xy, y y uv, u v is called the Jacobian of the coordinate transformation.- Example: What is the Jacobian for the transformation x rcos, y rsin? Example: Find the area bounded by the lines xy 1, xy, x3y and x3y 5. Use the substitution u x y and v x 3y. Example: Find the area bounded by the curves y x, y x, x y, and x 4y. 17

18 Substitutions in Triple Integrals Let x gu, v, w, y hu, v, w, z k u, v, w Then T,,,,,,,,,,,, f x y z dv f g u v w h u v w k u v w J u v w dudvdw D Where J u, v, w x x x u v w y y y x, y, z u v w u, v, w z z z u v w Example: Use the substitution x u, y v, z w to find the volume of the region bounded by x y z 1 and coordinate planes. J u, v, w 8 uvw so 1 1u 1uv 1 V dv 8uvwdwdvdu 90 T

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