triple integrals in cylindrical and spherical coordinates · mth 254 lesson 20. triple integrals in...

LESSON 20

Triple Integrals in Cylindrical andSpherical Coordinates

Contents20.1 Triple Integrals in Cylindrical Coordinates . . . . . . . . . . . . 2

20.2 Triple Integrals in Spherical Coordinates . . . . . . . . . . . . . 4

20.3 Applications of Triple Integrals . . . . . . . . . . . . . . . . . . . 7

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MTH 254 LESSON 20. TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES

20.1 Triple Integrals in Cylindrical Coordinates

Remember that we can find a double integral in polar coordinates using

�D

f(x, y) dA =

θf∫

θ0

ru(θ)∫

rl(θ)

f(r cos(θ), r sin(θ))r dr dθ

Now, in cylindrical coordinates we have

x = r cos(θ) y = r sin(θ) z = z

which is simply polar coordinates in x and y with the added z-component retaining its ownvalue. Thus, given a triple integral in rectangular coordinates, we can do the following:

�E

f(x, y, z) dV =�D

zu(x,y)∫

zl(x,y)

f(x, y, z) dz dA

=

θf∫

θ0

ru(θ)∫

rl(θ)

zu(r cos(θ),r sin(θ))∫

zl(r cos(θ),r sin(θ))

f(r cos(θ), r sin(θ), z) · r dz dr dθ

Essentially, we are able to evaluate a triple integral in cylindrical coordinates by convertingthe x and y-variables to polar coordinates just as we did with double integrals. Let’s putthis to practice:

Example 20.1.1 Evaluate�E

(x3 + xy2)dV where E is the solid in the first octant that

lies beneath the paraboloid z = 1 + x2 + y2 and within the cylinder x2 + y2 = 2.

Figure 20.1.1: The volume E to beintegrated overView in Geogebra:https://www.geogebra.org/3d/buvxpvjw

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Exercise 20.1.1 Use a triple integral to find the volume of the solid, E, which is boundedabove by the sphere x2 + y2 + z2 = 4 and below by the paraboloid 4z = 4− x2 − y2.

Figure 20.1.2: The volume V underx2 + y2 + z2 = 4 and above4z = 4− x2 − y2

View in Geogebra:https://www.geogebra.org/3d/djv3dy72

x

y

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20.2 Triple Integrals in Spherical Coordinates

In order to set up a triple integral in spherical coordinates with bounds in ρ, θ, and φ, weneed to slice the volume, E, into spherical wedges, find an approximating volume for eachslice, and then add the volumes up and then take the limit as the number of slices goes toinfinity. In order to find the volume of a single wedge in terms of Δρ, Δθ, and Δφ, let’sinvestigate figure

Here we see that the region Ei,j,k is ap-proximately a rectangular box. Whilethe opposite edges are not exactly thesame length, as we take the limit andmake these wedges smaller and smaller,the opposite edges limit out to be thesame length. Here we see the inner arcsto have dimensions ρiΔφ and riΔθ =ρi sin(φk)Δθ and the radial length to beΔρ.

Figure 20.2.1: Region Ei,j,k in SphericalCoordinates

Thus we find that the volume of Ei,j,k is approximately

V (Ei,j,k) ≈ ρiΔφ · ρi sin(φk)Δθ ·Δρ = ρ2i sin(φk)ΔρΔθΔφ

and therefore the hyper-volume given by�E

f(x, y, z) dV can be found in spherical coordi-

nates via�E

f(x, y, z) dV

= liml,m,n→∞

l∑i=1

m∑j=1

n∑k=1

f(ρi sin(φk) cos(θj), ρi sin(φk) sin(θj), ρi cos(φk)) ρ2i sin(φk)ΔρΔθΔφ

=

φf∫

φo

θu(φ)∫

θl(φ)

ρu(θ,φ)∫

ρl(θ,φ)

f(ρ sin(φ) cos(θ), ρ sin(φ) sin(θ), ρ cos(φ))ρ2 sin(φ)dρdθdφ

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Example 20.2.1 Evaluate�E

xyz dV where E lies between the spheres ρ = 2 and ρ = 4,

and above the cone φ = π3.

Figure 20.2.2: The regionE = {(ρ, θ, φ)|2 ≤ ρ ≤ 4, 0 ≤ φ ≤ π/3}.View in Geogebra:https://www.geogebra.org/3d/yerm4drv

x

y

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Exercise 20.2.1 Evaluate�E

x2 dV , where E is bounded by the xz-plane and the hemi-

spheres y =√9− x2 − z2 and y =

√16− x2 − z2.

Figure 20.2.3: The regionE = {(ρ, θ, φ)|3 ≤ ρ ≤ 4,−π ≤ θ ≤ 0}.View in Geogebra:https://www.geogebra.org/3d/uqba5bm7

x

y

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20.3 Applications of Triple Integrals

Example 20.3.1 Find the centroid of the solid that is bounded by the xz-plane and thehemispheres y =

√9− x2 − z2 and y =

√16− x2 − z2 assuming the density is constant.

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triple integrals in cylindrical and spherical coordinates · mth 254 lesson 20. triple integrals in...

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