Copyright © Cengage Learning. All rights reserved. 15.9 Triple Integrals in Spherical Coordinates2 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.3 Spherical Coordinates4 Spherical Coordinates The spherical coordinates (ρ, θ, φ) of a point P in space are shown in Figure 1, where ρ = |OP | is the distance from the origin to P, θ is the same angle as in cylindrical coordinates, and φ is the angle between the positive z-axis and the line segment OP. The spherical coordinates of a point Figure 15 Spherical Coordinates Note that ρ ≥ 0 0 ≤ φ ≤ π 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 ρ = c (see Figure 2); this is the reason for the name “spherical” coordinates. ρ = c, a sphere Figure 27 Spherical Coordinates The graph of the equation θ = c is a vertical half-plane (see Figure 3), and the equation φ = c represents a half-cone with the z-axis as its axis (see Figure 4). θ = c, a half-plane Figure 3 φ = c, a half-plane Figure 48 Spherical Coordinates The relationship between rectangular and spherical coordinates can be seen from Figure 5. From triangles OPQ and OPP ʹ′ we have z = ρ cos φ r = ρ sin φ Figure 59 Spherical Coordinates But x = r cos θ and y = r sin θ, 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 Example 1 The point (2, π /4, π /3) is given in spherical coordinates. Plot the point and find its rectangular coordinates. Solution: We plot the point in Figure 6. Figure 611 Example 1 – Solution From Equations 1 we have Thus the point (2, π /4, π /3) is in rectangular coordinates. cont’d12 Evaluating Triple Integrals with Spherical Coordinates13 Evaluating Triple Integrals with Spherical Coordinates In the spherical coordinate system the counterpart of a rectangular box is a spherical wedge E = {(ρ, θ, φ) | a ≤ ρ ≤ b, α ≤ θ ≤ β, c ≤ φ ≤ d } where a ≥ 0 and β – α ≤ 2π, and d – c ≤ π. 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 ρ = ρi, half-planes θ = θj, and half-cones φ = φk.14 Evaluating Triple Integrals with Spherical Coordinates Figure 7 shows that Eijk is approximately a rectangular box with dimensions Δρ, ρi Δ φ (arc of a circle with radius ρi, angle Δφ), and ρi sin φk Δ θ (arc of a circle with radius ρi sin φk, angle Δθ ). Figure 715 Evaluating Triple Integrals with Spherical Coordinates So an approximation to the volume of Eijk is given by ΔVijk ≈ (Δρ)(ρi Δφ)(ρi sin φk Δθ) = sin φk Δρ Δθ Δφ In fact, it can be shown, with the aid of the Mean Value Theorem that the volume of Eijk is given exactly by ΔVijk = sin φk Δρ Δθ Δφ where (ρi, θj, φk) is some point in Eijk.16 Evaluating Triple Integrals with Spherical Coordinates Let (xijk, yijk, zijk) be the rectangular coordinates of this point. Then17 Evaluating Triple Integrals with Spherical Coordinates But this sum is a Riemann sum for the function F(ρ, θ, φ ) = f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ ) ρ 2 sin φ Consequently, we have arrived at the following formula for triple integration in spherical coordinates.18 Evaluating Triple Integrals with Spherical Coordinates Formula 3 says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ using the appropriate limits of integration, and replacing dv by ρ 2 sin φ dρ dθ dφ.19 Evaluating Triple Integrals with Spherical Coordinates This is illustrated in Figure 8. Volume element in spherical coordinates: dV = ρ 2 sin φ dρ dθ dφ Figure 820 Evaluating Triple Integrals with Spherical Coordinates This formula can be extended to include more general spherical regions such as E = {(ρ, θ, φ) | α ≤ θ ≤ β, c ≤ φ ≤ d, g1(θ, φ) ≤ ρ ≤ g2(θ, φ)} In this case the formula is the same as in except that the limits of integration for ρ are g1(θ, φ ) and g2(θ, φ ). Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration.21 Example 4 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 922 Example 4 – 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 ρ 2 = ρ cos φ or ρ = ρ cos φ The equation of the cone can be written as23 Example 4 – Solution This gives sin φ = cos φ, or φ = π /4. Therefore the description of the solid E in spherical coordinates is E = {(ρ, θ, φ) | 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π/4, 0 ≤ ρ ≤ cos φ} cont’d24 Example 4 – Solution Figure 11 shows how E is swept out if we integrate first with respect to ρ, then φ, and then θ. Figure 11 φ varies from 0 to π /4 while θ is constant. ρ varies from 0 to cos φ while φ and θ are constant. θ varies from 0 to 2π . cont’d25 Example 4 – Solution The volume of E is
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