Section 15 8 Triple integrals in spherical coordinates MATH 251 Fall 2023 We have represented a point P in space as x y z rectangular coordinates and r z cylindrical coordinates In this lecture we learn spherical coordinates which are helpful in calculating triple integrals over regions bounded by spheres or cones Spherical coordinates of a point P are where is the distance from the origin to P is the same as in the cylindrical coordinates i e the angle between the projection of OP onto is the angle between the positive z axis and the line segment OP Here 0 the xy plane and the positive x axis Here 0 2 From spherical coordinates to rectangular coordinates x y z x sin cos y sin sin z cos From rectangular coordinates x y z to spherical coordinates px2 y2 z2 arccos z x 8 arctan y 6 is given in spherical coordinates Convert it to rectangular coordinates Example 1 The point 2 3 0 if the projection is in the rst quadrant if the projection is in the second or third quadrants 2 if the projection is in the fourth quadrant 4 5 1 2 Example 2 The point 0 1 p3 is given in rectangular coordinates Convert it to spherical coordi nates Example 3 Write the following equations in spherical coordinates a x2 y2 z2 16 b z px2 y2 Triple integrals in spherical coordinates Recall that the integration process involves dividing an object into smaller identical parts For example in Calculus 1 you divide an interval into subintervals of length dx in polar coordinates you divide the angle into angles d and the distance into lengths dr evaluating triple integrals in rectangular coordinates you form boxes of sides dx dy and dz Similarly in spherical coordinates you have d d and d How do we compute the volume of the below piece Recall that the arc length L corresponding to a sector of angle is L r 3 The volume is V i i sin k 2 This explains the following integration formula in spherical coordinates i sin k y E c Z f x y z dV Z d Z b a where E is the spherical wedge given by f sin cos sin sin cos 2 sin d d d E a b c d More generally when is bounded by functions g1 and g2 we have E g1 g2 c d and y E c Z f x y z dV Z d Z g2 g1 f sin cos sin sin cos 2 sin d d d 4 Example 4 Evaluate the integral by changing to spherical coordinates y E c Z f x y z dV Z d Z b 0 Z p9 x2 Z 3 a 0 Z p18 x2 y2 px2 y2 xdzdydx f sin cos sin sin cos 2 sin d d d Example 5 Evaluate tE xydV where E is within the sphere x2 y2 z2 1 and above the cone z px2 y2 5 6 Example 6 Consider the solid E that lies within the sphere x2 y2 z2 25 above the xy plane and inside the cylinder x2 y2 1 What is the ideal coordinate system rectangular cylindrical spherical we would use to nd the volume of E
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