15.9. Triple Integrals in Spherical Coordinates. Spherical Coordinates. Spherical Coordinates. Spherical Coordinates. Multiple Integrals

Transcription

1 15 Multiple Integrals 15.9 Triple Integrals in Spherical Coordinates Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Triple Integrals in Another useful coordinate system in three dimensions is the spherical coordinate system. It simplifies the evaluation of triple integrals over regions bounded by spheres or cones. 3 4 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. Note that 0 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. The spherical coordinates of a point Figure 1 5 6

2 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. 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 sphere Figure 2 = c, a half-plane Figure 3 = c, a half-plane Figure The relationship between rectangular and spherical coordinates can be seen from Figure 5. But x = r cos and y = r sin, so to convert from spherical to rectangular coordinates, we use the equations From triangles OPQ and OPP we have z = cos r = sin Also, the distance formula shows that We use this equation in converting from rectangular to spherical coordinates. Figure Example 1 Example 1 Solution The point (2, /4, /3) is given in spherical coordinates. Plot the point and find its rectangular coordinates. From Equations 1 we have Solution: We plot the point in Figure 6. Figure 6 11 Thus the point (2, /4, /3) is coordinates. in rectangular 12

3 Evaluating Triple Integrals with In the spherical coordinate system the counterpart of a rectangular box is a spherical wedge Evaluating Triple Integrals with 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. 13 So we divide E into smaller spherical wedges E ijk by means of equally spaced spheres = i, half-planes = j, and half-cones = k. 14 Evaluating Triple Integrals with Figure 7 shows that E ijk 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 ). Evaluating Triple Integrals with So an approximation to the volume of E ijk is given by V ijk ( )( i )( i sin k ) = sin k In fact, it can be shown, with the aid of the Mean Value Theorem that the volume of E ijk is given exactly by V ijk = sin k where ( i, j, k ) is some point in E ijk. Figure Evaluating Triple Integrals with Let (x ijk, y ijk, z ijk ) be the rectangular coordinates of this point. Then Evaluating Triple Integrals with 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

4 Evaluating Triple Integrals with Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing Evaluating Triple Integrals with This is illustrated in Figure 8. x = sin cos y = sin sin z = cos using the appropriate limits of integration, and replacing dv by 2 sin d d d. Volume element in spherical coordinates: dv = 2 sin d d d Figure Evaluating Triple Integrals with This formula can be extended to include more general spherical regions such as E = {(,, ), c d, g 1 (, ) g 2 (, )} Example 4 Use spherical coordinates to find the volume of the solid that lies above the cone and below the sphere x 2 + y 2 + z 2 = z. (See Figure 9.) In this case the formula is the same as in except that the limits of integration for are g 1 (, ) and g 2 (, ). Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration. Figure 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 This gives sin = cos, or = /4. Therefore the description of the solid E in spherical coordinates is E = {(,, ) 0 2, 0 /4, 0 cos } The equation of the cone can be written as 23 24

5 Figure 11 shows how E is swept out if we integrate first with respect to, then, and then. The volume of E is varies from 0 to cos while and are constant. varies from 0 to /4 while is constant. varies from 0 to 2. Figure