Triple Integrals Suppose we have a function f ( x, y, z ) defined on - - PDF document

Triple Integrals

Suppose we have a function f(x, y, z) defined on a solid S ⊂ R3 and we can estimate some quantity by n

i=1 f(x∗ i , y∗ i , z∗ i ) ∆Vi, where we

divide the solid into small, compact pieces of volume ∆Vi and choose a point (x∗

i , y∗ i , z∗ i ) in each piece.

We call those Riemann Sums. If they approach a limit, we call the limit the triple integral of f over the solid and denote it by

• S f(x, y, z) dV .

Applications of Triple Integrals

• S dV gives the volume of the solid S.
• If δ(x, y, z) is the density of the solid at the point (x, y, z), then

M =

• S δ(x, y, z) dV gives the mass of the solid.
• Myz =
• S xδ(x, y, z) dV is the moment about the yz-plane.
• Mxz =
• S yδ(x, y, z) dV is the moment about the xz-plane.
• Mxy =
• S zδ(x, y, z) dV is the moment about the xy-plane.
• If (x, y, z) is the center of mass of the solid, then x = Myz

M , y = Mxz M , z = Mxy M .

Evaluating a Triple Integral as an Iterated Integral

Suppose a solid S ⊂ R3 can be described as {(x, y, z)|α(x, y) ≤ z ≤ β(x, y), (x, y) ∈ D}, where D ⊂ R2 is a plane region. We can then evaluate

• S f(x, y, z) dS =
• D

β(x,y)

α(x,y) f(x, y, z) dz

• dA.

We may wish to write this in the form

• D dA

β(x,y)

α(x,y) dz f(x, y, z).

If D is a Type I region of the form {(x, y)|γ(x) ≤ y ≤ δ(x), a ≤ x ≤ b}, we may iterate the double integral to get

• S f(x, y, z) dV =

b

a dx

δ(x)

γ(x) dy

β(x,y)

α(x,y) dz f(x, y, z).

We may think of the solid as {(x, y, z)|α(x, y) ≤ z ≤ β(x, y), γ(x) ≤ y ≤ δ(x), a ≤ x ≤ b}.

Using Cylindrical Coordinates to Calculate Triple Integrals

Consider a small solid obtained by starting at a point (r, θ, z) and letting each of the coordinates increase by ∆r, ∆θ and ∆z. We get a solid which is almost a rectangular solid with vertices (r, θ, z), (r + ∆r, θ, z), (r, θ + ∆θ, z), (r, θ, z + ∆z), (r + ∆r, θ + ∆θ, z), (r + ∆r, θ, z + ∆z), (r, θ + ∆θ, z + ∆z) and (r + ∆r, θ + ∆θ, z + ∆z). The edge from (r, θ, z) to (r + ∆r, θ, z) will have length ∆r.

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The edge from (r, θ, z) to (r, θ + ∆θ, z) will have length r∆θ since it is parallel to the arc of a sector of a circle in the rθ−plane with radius r and angle ∆θ. The edge from (r, θ, z) to (r, θ, z + ∆z) will have length ∆z. Thus the volume of the small solid will be ∆V ≈ (∆r)(r∆θ)(∆z) = r∆r∆θ∆z.

Cylindrical Coordinates

A sum n

i=1 f(r∗ i , θ∗ i , z∗ i )∆Vi will thus ≈ n i=1 f(r∗ i , θ∗ i , z∗ i )r∗ i ∆ri ∆θi ∆zi

and we may conclude

• S f(r, θ, z) dV =
• S f(r, θ, z) r dr dθ dz.

Using Spherical Coordinates to Evaluate Triple Integrals

Consider a small solid obtained by starting at a point (ρ, θ, φ) and letting each of the coordinates increase by ∆ρ, ∆θ and ∆φ. We get a solid which is almost a rectangular solid with vertices (ρ, θ, φ), (ρ + ∆ρ, θ, φ), (ρ, θ + ∆θ, φ), (ρ, θ, φ + ∆φ), (ρ + ∆ρ, θ + ∆θ, φ), (ρ + ∆ρ, θ, φ + ∆φ), (ρ, θ + ∆θ, φ + ∆φ) and (ρ + ∆ρ, θ + ∆θ, φ + ∆φ).

Spherical Coordinates

The edge from (ρ, θ, φ) to (ρ + ∆ρ, θ, φ) will have length ∆ρ. The edge from (ρ, θ, φ) to (ρ, θ+∆θ, φ) will have length r∆θ, where r is the corresponding cylindrical coordinate of the point. This was shown

when looking at cylindrical coordinates. Since r = ρ sin φ, the edge will

have length ρ sin φ∆θ. The edge from (ρ, θ, φ) to (ρ, θ, φ+∆φ) will have length ρ∆φ, since it’s an arc subtended by an angle ∆φ in a circle of radius ρ. Thus the volume of the small solid will be ∆V ≈ (∆ρ)(ρ sin φ∆θ)(ρ∆φ) = ρ2 sin φ∆ρ∆θ∆φ. A sum n

i=1 f(ρ∗ i , θ∗ i , φ∗ i )∆Vi will thus be

≈ n

i=1 f(ρ∗ i , θ∗ i , φ∗ i )(ρ∗ i )2 sin(φ∗ i )∆ρi ∆θi ∆φi and we may conclude

• S f(ρ, θ, φ) dV =
• S f(ρ, θ, φ) ρ2 sin φ dρ dθ dφ.

Change of Variable

Using polar, cylindrical or spherical coordinates are special cases of a more general technique of a transformation, mapping or change of

• variables. It is a generalization of a change of variable for an ordinary

integral. Consider a transformation T(u, v) = (x, y), where x = g(u, v), y = h(u, v) for some functions g : R2 → R and h : R2 → R, which associates

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with every point (u, v) in some region S a corresponding point (x, y) in a region R. Assume the transformation is 1 − 1, onto and C1, meaning that g and h have continuous first-order partial derivatives. Suppose we take a small rectangle with vertices (u0, v0), (u0 + ∆u, v0), (u0, v0 + ∆v), (u0 + ∆u, v0 + ∆v). It will map into a region which is almost a parallelogram. Let T(u0, v0) = (x0, y0). T(u0 + ∆u, v0) = (f(u0 + ∆u, v0), g(u0 + ∆u, v0)). Using differentials, f(u0 + ∆u, v0) ≈ x0 + ∂x ∂u∆u. Similarly, g(u0 + ∆u, v0) ≈ y0 + ∂x ∂u∆u. Effectively, one side of the parallelogram is the vector α =< ∂x ∂u∆u, ∂y ∂u∆u >=< ∂x ∂u, ∂y ∂u > ∆u. Similarly, the other side of the parallelogram is effectively the vector β =< ∂x ∂v ∆v, ∂y ∂v∆v >=< ∂x ∂v , ∂y ∂v > ∆v. The area of the parallelogram is |α||β| sin θ, where θ is the angle be- tween the vectors. If we embed the two vectors in R3 by adding a third component of 0 to each, we can calculate that with the cross product. < ∂x ∂u, ∂y ∂u, 0 > ∆u× < ∂x ∂v , ∂y ∂v, 0 > ∆v =

• i

j k ∂x ∂u ∂y ∂u ∂x ∂v ∂y ∂v

• ∆u ∆v =
• ∂x

∂u ∂y ∂u ∂x ∂v ∂y ∂v

• ∆u ∆v k.

The length of the cross product is thus

• ∂(x, y)

∂(u, v)

• ∆u ∆v, where ∂(x, y)

∂(u, v) =

• ∂x

∂u ∂y ∂u ∂x ∂v ∂y ∂v

• is called the Jacobian of the transformation.

Thus, the area of the image in the xy−plane of the rectangle in the uv−plane is approximately

• ∂(x, y)

∂(u, v)

• ∆u ∆v.

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If we had a Riemann sum n

i=1 f(xi, yi) ∆Ai over the region R, we

could approximate it by n

i=1 f(g(ui, vi), h(ui, vi))

• ∂(x, y)

∂(u, v)

• ∆ui ∆vi.

We conclude

• R f(x, y) dA =
• S f(g(u, v), h(u, v))
• ∂(x, y)

∂(u, v)

• du dv,

which can be written as

• S f(x(u, v), y(u, v))
• ∂(x, y)

∂(u, v)

• du dv.

Change of Variables in Higher Dimensions

In higher dimensions, the visualization is trickier but the analogous results hold. For example, in R3, we would get

• R f(x, y) dV =
• S f(x(u, v, w), y(u, v, w), z(u, v, w))
• ∂(x, y, z)

∂(u, v, w)

• du dv dw,

where ∂(x, y, z) ∂(u, v, w) =

• ∂x

∂u ∂y ∂u ∂z ∂u ∂x ∂v ∂y ∂v ∂z ∂v ∂x ∂w ∂y ∂w ∂z ∂w

• .