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TRIPLE INTEGRALS IN SPHERICAL COORDINATES
SECTION 12.7 TRIPLE INTEGRALS IN SPHERICAL COORDINATES
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SPHERICAL COORDINATE SYSTEM
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. 12.7
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SPHERICAL COORDINATES
The spherical coordinates (r, , f) of a point P in space are shown in Figure 1. r = |OP| is the distance from the origin to P. is the same angle as in cylindrical coordinates. f is the angle between the positive z-axis and the line segment OP. 12.7
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SPHERICAL COORDINATES
Note that: r ≥ 0 0 ≤ ≤ π 12.7
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SPHERICAL COORDINATE SYSTEM
The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. 12.7
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SPHERE For example, the sphere with center the origin and radius c has the simple equation r = c. This is the reason for the name “spherical” coordinates. See Figure 2. 12.7
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The graph of the equation = c is a vertical half-plane.
See Figure 3. 12.7
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The equation f = c represents a half-cone with the z-axis as its axis.
See Figure 4. 12.7
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SPHERICAL & RECTANGULAR COORDINATES
The relationship between rectangular and spherical coordinates can be seen from Figure 5. 12.7
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SPHERICAL & RECTANGULAR COORDINATES
From triangles OPQ and OPP’, we have: z = r cos f r = r sin f However, x = r cos y = r sin 12.7
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SPH. & RECT. COORDINATES So, to convert from spherical to rectangular coordinates, we use the equations x = r sin f cos y = r sin f sin z = r cos f 12.7
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Also, the distance formula shows that: r 2 = x2 + y2 + z2
SPH. & RECT. COORDINATES Also, the distance formula shows that: r 2 = x2 + y2 + z2 We use this equation in converting from rectangular to spherical coordinates. 12.7
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The point (2, p/4, p /3) is given in spherical coordinates.
Example 1 The point (2, p/4, p /3) is given in spherical coordinates. Plot the point and find its rectangular coordinates. 12.7
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We plot the point in Figure 6.
Example 1 SOLUTION We plot the point in Figure 6. 12.7
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From Equations 1, we have:
Example 1 SOLUTION From Equations 1, we have: 12.7
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Thus, the point (2, p/4, p /3) is in rectangular coordinates.
Example 1 SOLUTION Thus, the point (2, p/4, p /3) is in rectangular coordinates. 12.7
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The point is given in rectangular coordinates.
Example 2 The point is given in rectangular coordinates. Find spherical coordinates for the point. 12.7
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Example 2 SOLUTION From Equation 2, we have: 12.7
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Therefore, spherical coordinates of the given point are:
Example 2 SOLUTION So, Equations 1 give: Note that ≠ 3p/2 because Therefore, spherical coordinates of the given point are: (4, p/2 , 2p/3) 12.7
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EVALUATING TRIPLE INTEGRALS WITH SPHERICAL COORDINATES
In the spherical coordinate system, the counterpart of a rectangular box is a spherical wedge where a ≥ 0, b – a ≤ 2p, d – c ≤ p 12.7
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EVALUATING TRIPLE INTEGRALS
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. 12.7
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EVALUATING TRIPLE INTEGRALS
Thus, we divide E into smaller spherical wedges Eijk by means of equally spaced spheres r = ri, half-planes = j, and half-cones f = fk. 12.7
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EVALUATING TRIPLE INTEGRALS
Figure 7 shows that Eijk is approximately a rectangular box with dimensions Δr : ri Δfk (arc of a circle with radius ri, angle Δfk) ri sinfkΔj (arc of a circle with radius ri sin fk, angle Δj) 12.7
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EVALUATING TRIPLE INTEGRALS
So, an approximation to the volume of Eijk is given by: 12.7
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EVALUATING TRIPLE INTEGRALS
In fact, it can be shown, with the aid of the Mean Value Theorem, that the volume of Eijk is given exactly by: where is some point in Eijk. 12.7
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EVALUATING TRIPLE INTEGRALS
Let be the rectangular coordinates of that point. 12.7
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EVALUATING TRIPLE INTEGRALS
Then, 12.7
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EVALUATING TRIPLE INTEGRALS
However, that sum is a Riemann sum for the function So, we have arrived at the following formula for triple integration in spherical coordinates. 12.7
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where E is a spherical wedge given by:
Formula 3 where E is a spherical wedge given by: 12.7
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TRIPLE INTEGRALS WITH SPHERICAL COORDINATES
Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing: x = r sin f cos y = r sin f sin z = r cos f 12.7
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TRIPLE INTEGRALS WITH SPHERICAL COORDINATES
That is done by: Using the appropriate limits of integration. Replacing dV by r2 sinf dr d df. This is illustrated in Figure 8. 12.7
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TRIPLE INTEGRALS WITH SPHERICAL COORDINATES
The formula can be extended to include more general spherical regions such as: The formula is the same as in Formula 3 except that the limits of integration for r are g1(, f) and g2(, f). 12.7
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TRIPLE INTEGRALS WITH SPHERICAL COORDINATES
Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration. 12.7
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Evaluate where B is the unit ball
Example 3 Evaluate where B is the unit ball 12.7
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As the boundary of B is a sphere, we use spherical coordinates:
Example 3 SOLUTION As the boundary of B is a sphere, we use spherical coordinates: In addition, spherical coordinates are appropriate because: x2 + y2 + z2 = r2 12.7
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Example 3 SOLUTION So, Formula 3 gives: 12.7
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Note It would have been extremely awkward to evaluate the integral in Example 3 without spherical coordinates. In rectangular coordinates, the iterated integral would have been 12.7
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Use spherical coordinates to find the volume of the solid that lies:
Example 4 Use spherical coordinates to find the volume of the solid that lies: Above the cone Below the sphere x2 + y2 + z2 = z See Figure 9. 12.7
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Example 4 SOLUTION Notice that the sphere passes through the origin and has center (0, 0, ½). We write its equation in spherical coordinates as: r2 = r cos f or r = cos f 12.7
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The equation of the cone can be written as:
Example 4 SOLUTION The equation of the cone can be written as: This gives: sin f = cos f or f = p/4 12.7
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Thus, the description of the solid E in spherical coordinates is:
Example 4 SOLUTION Thus, the description of the solid E in spherical coordinates is: 12.7
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Example 4 SOLUTION Figure 11 shows how E is swept out if we integrate first with respect to r, then f, and then . 12.7
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Example 4 SOLUTION The volume of E is: 12.7
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TRIPLE INTEGRALS WITH SPHERICAL COORDINATES
Figure 10 gives another look (this time drawn by Maple) at the solid of Example 4. 12.7
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