Integration

Total Mass of a Region in 3-Space 1D 2D Cylindrical Coordinates Spherical Coordinates Contents

In order to understand how integration works in 3-space, consider the problem of finding the total mass of some region.

If we know density as a function f of x, y, and z, then we can approximate the mass of a tiny box of volume dV located at (x₀, y₀, z₀) as the product f(x₀, y₀, z₀)dV. We can then find the total mass by adding up these the masses of these tiny volumes.

If all three dimensions of the volumes approach 0, the summation becomes a triple integral.

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Exercises

How could the total mass method be used as a way to understand integration for functions of one or two variables?

Triple and Repeated Riemann Integrals 1D 2D Top of Page Contents

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Exercises

Riemann integrals for functions of one variable can use left- or right- hand sums, lower or upper sums, and trapezoidal approximations. Describe some corresponding methods for integrals of functions of 3 variables.

Total Mass of Regions between Function Graphs 1D 2D Cylindrical Coordinates Spherical Coordinates Top of Page Contents

As was true with simple regions, in order to find the total mass of a region between two function graphs, we must integrate the density function times some tiny volume dV over the given region. The difference here is in the limits of integration. x and y will still begin and end at constants. The lower and upper limits of z, however, will be functions of x and y.

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Exercises

In what ways does this lab parallel the "Volume between Function Graphs" lab? In what ways is it different?

Change of Order of Integration 2D Cylindrical Coordinates Spherical Coordinates Top of Page Contents

Applied to continuous functions of three variables, Fubini's Theorem says that there are six different ways to evaluate integrals.

For a function f(x, y, z) with domain R such that a ≤ x ≤ b, c ≤ y ≤ d, k ≤ z ≤ l,

∫_a^b∫_c^d∫_k^lf(x, y, z)dzdydx =

∫_a^b∫_k^l∫_c^df(x, y, z)dydzdx =

∫_c^d∫_a^b∫_k^lf(x, y, z)dzdxdy =

∫_c^d∫_k^l∫_a^bf(x, y, z)dxdzdy =

∫_k^l∫_a^b∫_c^df(x, y, z)dydxdz =

∫_k^l∫_c^d∫_a^bf(x, y, z)dxdydz =

∫∫∫_Rf(x, y, z)dV

[D]

Exercises

In the 2-variable lab on Change of Order of Integration, there is a demo called "Slab Approximations." What would be the equivalent of a slab for integrals over three variables?

Change of Variables 1D 2D Top of Page Contents

Integration in a non-Cartesian coordinate system requires an application of the Change of Variables Theorem.

Consider what would happen for a function of three variables. Up until now, we have integrated over three variables by adding up the masses of rectangular prisms of density f(x, y, z) and volume dx * dy * dz. If we decide to express x, y, and z as functions of variables u, v, and w, we need to find some new expression for the volume of the differential.

The density will still be the value of the function, expressed as f(x(u,v,w), y(u,v,w), z(u,v,w)).

The tiny volume will be a parallelepiped. The three vectors representing the edges of this parallelepiped are based on what happens when u, v, and w change. When u changes by some amount du, x increases by x_u*du, y increases by y_u*du, and z increases by z_u*du. The resulting vector is (x_u * du, y_u * du, z_u*du).

The other two vectors, which are derived similarly, based on what happens when v increases and when w increases, are (x_v * dv, y_v * dv, z_v * dv) and (x_w * dw, y_w * dw, z_w * dw).

To find the volume of this parallelepiped, take the dot product of one edge vector with the cross product of the two other edge vectors. For example, you could take the cross product of the vectors (x_v * dv, y_v * dv, z_v*dv) and (x_w * dw, y_w * dw, z_w*dw):

(y_vz_w - y_wz_v, z_vx_w - z_wx_v, x_vy_w - x_wy_v)dudv.

Now take the dot product of this vector with the remaining edge vector (x_u * du, y_u * du, z_u * du):

(x_u[y_vz_w - y_wz_v] + y_u[z_vx_w - z_wx_v] + z_u[x_vy_w - x_wy_v])dudvdw.

Finally, multiply by density to get the Change of Variables Theorem for triple integrals:

∫∫∫_Df(x, y, z)dxdydz = ∫∫∫_D*f(x(u, v, w), y(u, v, w), z(u, v, w))(x_u[y_vz_w - y_wz_v] + y_u[z_vx_w - z_wx_v] + z_u[x_vy_w - x_wy_v])dudvdw,

Where D is the domain in Cartesian coordinates and D* is the domain in the new coordinate system.

Note that the expression

(x_u[y_vz_w - y_wz_v] + y_u[z_vx_w - z_wx_v] + z_u[x_vy_w - x_wy_v])

is equivalent to the determinant of the matrix with rows

(x_u, x_v, x_w), (y_u, y_v, y_w), (z_u, z_v, z_w).

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Exercises

1. Find the change of variables formula (i.e. form of

∫∫∫_Df(x, y, z)dxdydz = ∫∫∫_D*f(x(u, v, w), y(u, v, w), z(u, v, w))(x_u[y_vz_w - y_wz_v] + y_u[z_vx_w - z_wx_v] + z_u[x_vy_w - x_wy_v])dudvdw) for each of the following coordinate systems:

x(u, v, w) = (u - v), y(u, v, w) = (u + v), z(u, v, w) = (w + u)
x(u, v, w) = ucos(v), y(u, v, w) = usin(v), z(u, v, w) = w (cylindrical coordinates)
x(u, v, w) = ucos(v)cos(w), y(u, v, w) = usin(v)cos(w), z(u, v, w) = sin(w) (spherical coordinates)

2. Under what conditions will the cyan parallelepiped shown in the "D" window have constant volume for all (u, v, w)?

3. How does the change of variables formula for three variables simplify for the case that z = w, and x and y are functions of u and v? Find an example of a situation where this simplification would apply.

Center of Mass 1D 2D Cylindrical Coordinates Spherical Coordinates Top of Page Contents

The center of mass of a three-dimensional region is a weighted average of the positions of the particles that the region comprises. The amount each particle counts in this weighted average is proportional to its mass.

If we wanted to find the average position of a series of particles, we could find each coordinate of the point representing average position by adding up the values of that coordinate from each point and dividing by the number of points. For example, to find the x coordinate X in the average position of n particles, we would calculate

(Σ_nx_n)/n

Where x_n is the x coordinate of the position of the nth particle.

Suppose instead we wanted to find the center of mass of a region as described in the first paragraph. Our expression for the x coordinate of the center of mass, given the mass m_n of each particle is:

(Σ_nx_nm_n)/(Σ_nm_n)

(Note that the denominator is equal to the total mass of the set of particles.)

Given a density function ρ(x, y, z), we can express this summation as

(Σ_nx_n*ρ(x_n, y_n, z_n)*A_n)/(Σ_nρ(x_n, y_n, z_n)*V_n)

Where V_n is the volume of the nth particle.

For a region with domain D composed of a very large number of particles, we can change this summation to a double integral, replacing V_n with dV:

X = (∫∫∫_Dx*ρ(x, y, z)dV)/(∫∫∫_Dρ(x, y, z)dV).

Similarly, to find the y coordinate of center of mass, use:

Y = (∫∫∫_Dy*ρ(x, y, z)dV)/(∫∫∫_Dρ(x, y, z)dV)

and to find the z coordinate, use

Z = (∫∫∫_Dz*ρ(x, y, z)dV)/(∫∫∫_Dρ(x, y, z)dV).

This gives us the center of mass (X, Y, Z) of D.

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[D]

Exercises

1. Does the center of mass change if the density function is multipied by some constant k? Why or why not?

2. Find a region and some density function ρ(x, y, z) such that the center of mass lies outside the region. In general, what geometric property must a region have for its center of mass to be able to lie outside of the region?

Moment of Intertia 2D Cylindrical Coordinates Spherical Coordinates Top of Page Contents

The moment of inertia I of an object about an axis of rotation is a value which indicates the resistance of that object to changes in rotation about that axis. For a three-dimensional region D with density ρ(x, y), this calculated using the following formula:

I = ∫(r(x, y, z))²ρ(x, y, z)dxdydz

Where r(x, y, z) is the distance of the point (x, y, z) from the axis of rotation.

For an object of density 1, the moment of inertia will be the density multiplied by the integral

I = ∫(r(x, y, z))²dxdydz.

[D]

Exercises

1. Find the moment of inertia of a sphere of radius 1 and uniform density 1 rotated about an axis which passes through its center.

2. Find the moment of inertia of a sphere of radius R, uniform density, and total mass M rotated about an axis which passes through its center.

3. Find the moment of inertia of a cylinder of radius 1 and uniform density 1 rotated about an axis which passes through its center, such that the cylinder rotates as a wheel would rotate.

4. Find the moment of inertia of a cylinder of radius R, uniform density, and total mass M rotated about an axis which passes through its center, such that the cylinder rotates as a wheel would rotate.

5. Compare your result from question 4. to the moment of inertia of a circle of radius R, uniform density, and total mass M rotated about its center.