15 Multiple Integrals Copyright Cengage Learning All rights reserved 15 9 Triple Integrals in Spherical Coordinates Copyright Cengage Learning All rights reserved Triple Integrals in Spherical Coordinates Learning objectives convert between rectangular and spherical coordinates evaluate triple integrals with spherical coordinates 3 Spherical Coordinates 4 Spherical Coordinates The spherical coordinates of a point P in space are shown in Figure 1 where OP is the distance from the origin to P is the same angle as in cylindrical coordinates and is the angle between the positive z axis and the line segment OP The spherical coordinates of a point Figure 1 5 Spherical Coordinates Note that 0 0 2 0 The spherical coordinate system is especially useful in problems where there is symmetry about a point and the origin is placed at this point 6 Spherical Coordinates For example the sphere with center the origin and radius c has the simple equation c see Figure 2 this is the reason for the name spherical coordinates c a sphere Figure 2 7 Spherical Coordinates The graph of the equation c is a vertical half plane see Figure 3 and the equation c represents a half cone with the z axis as its axis see Figure 4 c a half plane Figure 3 c a half plane Figure 4 8 Spherical Coordinates The relationship between rectangular and spherical coordinates can be seen from Figure 5 From triangles OPQ and OPP we have z cos r sin Figure 5 9 Spherical Coordinates But x r cos and y r sin so to convert from spherical to rectangular coordinates we use the equations Also the distance formula shows that We use this equation in converting from rectangular to spherical coordinates 10 Evaluating Triple Integrals with Spherical Coordinates 11 Evaluating Triple Integrals with Spherical Coordinates In the spherical coordinate system the counterpart of a rectangular box is a spherical wedge E a b c d where a 0 and 2 and d c Although we defined triple integrals by dividing solids into small boxes it can be shown that dividing a solid into small spherical wedges always gives the same result So we divide E into smaller spherical wedges Eijk by means of equally spaced spheres i half planes j and half cones k 12 Evaluating Triple Integrals with Spherical Coordinates Figure 7 shows that Eijk is approximately a rectangular box with dimensions i arc of a circle with radius i angle and i sin k arc of a circle with radius i sin k angle Figure 7 13 Evaluating Triple Integrals with Spherical Coordinates So an approximation to the volume of Eijk is given by Vijk i i sin k sin k In fact it can be shown with the aid of the Mean Value Theorem that the volume of Eijk is given exactly by Vijk sin k where i j k is some point in Eijk 14 Evaluating Triple Integrals with Spherical Coordinates Let xijk yijk zijk be the rectangular coordinates of this point Then 15 Evaluating Triple Integrals with Spherical Coordinates But this sum is a Riemann sum for the function F f sin cos sin sin cos 2 sin Consequently we have arrived at the following formula for triple integration in spherical coordinates 16 Evaluating Triple Integrals with Spherical Coordinates Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing x sin cos y sin sin z cos using the appropriate limits of integration and replacing dv by 2 sin d d d 17 Evaluating Triple Integrals with Spherical Coordinates This is illustrated in Figure 8 Volume element in spherical coordinates dV 2 sin d d d Figure 8 18 Evaluating Triple Integrals with Spherical Coordinates This formula can be extended to include more general spherical regions such as E c d g1 g2 In this case the formula is the same as in except that the limits of integration for are g1 and g2 Usually spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration 19 Example Use spherical coordinates to find the volume of the solid that lies above the cone and below the sphere x2 y2 z2 z See Figure 9 Figure 9 20 Example Solution Notice that the sphere passes through the origin and has center 0 0 We write the equation of the sphere in spherical coordinates as 2 cos or cos The equation of the cone can be written as 21 Example Solution continued This gives sin cos or 4 Therefore the description of the solid E in spherical coordinates is E 0 2 0 4 0 cos 22 Example Solution continued Figure 11 shows how E is swept out if we integrate first with respect to then and then varies from 0 to cos while and are constant varies from 0 to 4 while is constant varies from 0 to 2 Figure 11 23 Example Solution continued The volume of E is 24
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