Stewart J. - 15.8 - Coordenadas Esfericas

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SECTION 15.8 TRIPLE INTEGRALS IN SPHERICAL COORDINATES

D I S COV E RY PROJECT

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1005

THE INTERSECTION OF THREE CYLINDERS The figure shows the solid enclosed by three circular cylinders with the same diameter that intersect at right angles. In this project we compute its volume and determine how its shape changes if the cylinders have different diameters.

1. Sketch carefully the solid enclosed by the three cylinders x 2 y 2 苷 1, x 2 z 2 苷 1, and

y 2 z 2 苷 1. Indicate the positions of the coordinate axes and label the faces with the equations of the corresponding cylinders.

2. Find the volume of the solid in Problem 1. CAS

3. Use a computer algebra system to draw the edges of the solid. 4. What happens to the solid in Problem 1 if the radius of the first cylinder is different from 1?

Illustrate with a hand-drawn sketch or a computer graph. 5. If the first cylinder is x 2 y 2 苷 a 2, where a 1, set up, but do not evaluate, a double inte-

gral for the volume of the solid. What if a 1?

15.8

TRIPLE INTEGRALS IN SPHERICAL COORDINATES 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. SPHERIC AL COORDINATES

The spherical coordinates 共 , , 兲 of a point P in space are shown in Figure 1, where 苷 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

ⱍ

ⱍ

0

0

z P( ∏, ¨, ˙)

∏ ˙ O

¨ FIGURE 1

The spherical coordinates of a point

x

y

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CHAPTER 15 MULTIPLE INTEGRALS

The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. For example, the sphere with center the origin and radius c has the simple equation 苡 c (see Figure 2); this is the reason for the name “sphericalâ€? coordinates. The graph of the equation 苡 c is a vertical half-plane (see Figure 3), and the equation 苡 c represents a half-cone with the z-axis as its axis (see Figure 4). z

z

z

z

c 0

0

0

c

y x

0 y

x

y

y

x

x

Ď€/2<c<Ď€

0<c<Ď€/2 FIGURE 2 âˆ?=c, a sphere

FIGURE 3 ¨=c, a half-plane

z

z 苡 cos

P(x, y, z) P(âˆ?, ¨, Ë™)

z

�

O 1

r

¨ y

r 苡 sin

But x 苡 r cos and y 苡 r sin , so to convert from spherical to rectangular coordinates, we use the equations

Ë™

Ë™

x

FIGURE 4 Ë™=c, a half-cone

The relationship between rectangular and spherical coordinates can be seen from Figure 5. From triangles OPQ and OPP we have

Q

x

c

x 苡 sin cos

y 苡 sin sin

z 苡 cos

y P ª(x, y, 0)

Also, the distance formula shows that

FIGURE 5

2 苡 x 2 y 2 z2

2

We use this equation in converting from rectangular to spherical coordinates. V EXAMPLE 1 The point ĺ…ą2, ĺ…ž4, ĺ…ž3ĺ…˛ is given in spherical coordinates. Plot the point and find its rectangular coordinates.

SOLUTION We plot the point in Figure 6. From Equations 1 we have z

s3 cos 苡2 3 4 2

y 苡 sin sin 苡 2 sin

s3 sin 苡2 3 4 2

(2, Ď€/4, Ď€/3) Ď€ 3

2

O

x

Ď€ 4

FIGURE 6

y

冉 冊冉 冊 冑 冉 冊冉 冊 冑

x 苡 sin cos 苡 2 sin

z 苡 cos 苡 2 cos

1 s2

1 s2

苡

苡

3 2

3 2

苡 2( 12 ) 苡 1 3

Thus the point ĺ…ą2, ĺ…ž4, ĺ…ž3ĺ…˛ is (s3ĺ…ž2 , s3ĺ…ž2 , 1) in rectangular coordinates.

M

SECTION 15.8 TRIPLE INTEGRALS IN SPHERICAL COORDINATES

| WARNING There is not universal agreement on the notation for spherical coordinates. Most books on physics reverse the meanings of and and use r in place of .

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The point (0, 2s3 , 2) is given in rectangular coordinates. Find spherical coordinates for this point. V EXAMPLE 2

SOLUTION From Equation 2 we have

苡 sx 2 y 2 z 2 苡 s0 12 4 苡 4 and so Equations 1 give

TEC In Module 15.8 you can investigate families of surfaces in cylindrical and spherical coordinates.

cos 苡

z 2 1 苡 苡 4 2

苡

2 3

cos 苡

x 苡0 sin

苡

2

(Note that 苡 3 兞2 because y 苡 2s3 given point are 兹4, 兞2, 2 兞3兲.

0.) Therefore spherical coordinates of the M

EVALUATING TRIPLE INTEGRALS WITH SPHERIC AL COORDINATES

In the spherical coordinate system the counterpart of a rectangular box is a spherical wedge E 苡 兾 兹 , , 兲 a b, , c d 兜

âą?

z

âˆ? i sin Ë™ k ĂŽ¨

˙k

ĂŽË™ âˆ? i  ĂŽË™

0

x

where a 0, 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 苡 i , half-planes 苡 j , and half-cones 苡 k . 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 ). So an approximation to the volume of Eijk is given by

ĂŽâˆ?

ri=âˆ? i sin Ë™ k

ĂŽ¨

y

ri  ĂŽ¨=âˆ? i sin Ë™ k ĂŽ¨ FIGURE 7

Vijk ⏇ 兹 兲兹 i 兲兹 i sin k 兲 苡 2i sin k In fact, it can be shown, with the aid of the Mean Value Theorem (Exercise 45), that the volume of Eijk is given exactly by 苲

Vijk 苡 苲 i2 sin k 苲

苲

* , y ijk * , z ijk * 兲 be the rectangular coordinates of where 兹 苲 i , j , k 兲 is some point in Eijk . Let 兹x ijk this point. Then l

yyy f 兹x, y, z兲 dV 苡 E l

苡 lim

lim

m

n

ĺ…ş ĺ…ş ĺ…ş f ĺ…ąx * , y * , z * ĺ…˛ V

l, m, n l i苡1 j苡1 k苡1 m

n

ĺ…ş ĺ…ş ĺ…ş f ĺ…ą

l, m, n l i苡1 j苡1 k苡1

苲

苲

i

ijk

ijk

ijk

苲

ijk

苲

苲

苲

苲

sin k cos j, 苲 i sin k sin j , 苲 i cos k 兲 苲 i2 sin k

But this sum is a Riemann sum for the function F兹 , , 兲 苡 f 兹 sin cos , sin sin , cos 兲 2 sin Consequently, we have arrived at the following formula for triple integration in spherical coordinates.

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