The Main Idea 

When setting up triple integrals in cylindrical coordinates, the critical element is the volume element [latex]dV = r \, dr \, d\theta \, dz[/latex]. That extra factor of [latex]r[/latex] is essential—it comes from the geometry of how cylindrical “wedges” expand as you move away from the [latex]z[/latex]-axis.

Converting to Cylindrical Coordinates

To convert a triple integral from rectangular to cylindrical coordinates, substitute [latex]x = r\cos\theta[/latex], [latex]y = r\sin\theta[/latex], and [latex]z = z[/latex] into the function. The integral becomes:

[latex]\underset{B}{\iiint} g(x,y,z) , dV = \underset{B}{\iiint} g(r\cos\theta, r\sin\theta, z) \, r \, dr \, d\theta \, dz[/latex]

Don’t forget that extra [latex]r[/latex]! It’s not part of the function—it’s part of the volume element itself.

Common Surface Equations

Cylindrical coordinates simplify many surface equations. A circular cylinder [latex]x^2 + y^2 = c^2[/latex] becomes simply [latex]r = c[/latex]. A cone [latex]z^2 = c^2(x^2 + y^2)[/latex] becomes [latex]z = cr[/latex]. A paraboloid [latex]z = c(x^2 + y^2)[/latex] becomes [latex]z = cr^2[/latex]. A sphere [latex]x^2 + y^2 + z^2 = c^2[/latex] becomes [latex]r^2 + z^2 = c^2[/latex].

Setting Up Bounds for General Regions

For a general region [latex]E[/latex] bounded between two surfaces, project onto the [latex]r\theta[/latex]-plane to find [latex]D[/latex]. If [latex]z[/latex] varies between [latex]u_1(r,\theta)[/latex] and [latex]u_2(r,\theta)[/latex], the integral becomes:

[latex]\underset{E}{\iiint} f(r,\theta,z) \, r \, dr \, d\theta \, dz = \underset{D}{\iint} \left[\int_{u_1(r,\theta)}^{u_2(r,\theta)} f(r,\theta,z) , dz\right] r \, dr \, d\theta[/latex]

Problem-Solving Strategy

1. Identify surfaces with circular symmetry or expressions containing [latex]x^2 + y^2[/latex]
2. Convert surface equations to cylindrical form
3. Determine bounds by examining projections and intersections
4. Set up the integral with the volume element [latex]r \, dr \, d\theta \, dz[/latex]
5. Integrate in the order that simplifies computation—the order can be changed just as with rectangular coordinates

The Main Idea 

Spherical coordinates shine when dealing with regions that have symmetry around a point, especially spheres, cones, and other shapes centered at the origin. The critical element is the volume element [latex]dV = \rho^2 \sin\varphi , d\rho , d\varphi , d\theta[/latex]. Those extra factors [latex]\rho^2 \sin\varphi[/latex] account for how spherical “wedges” change size based on distance from the origin and angle from the [latex]z[/latex]-axis.

Converting to Spherical Coordinates

To convert a triple integral from rectangular to spherical coordinates, substitute [latex]x = \rho\sin\varphi\cos\theta[/latex], [latex]y = \rho\sin\varphi\sin\theta[/latex], and [latex]z = \rho\cos\varphi[/latex] into the function. The integral becomes:

[latex]\underset{B}{\iiint} g(x,y,z) \, dV = \underset{B}{\iiint} g(\rho\sin\varphi\cos\theta, \rho\sin\varphi\sin\theta, \rho\cos\varphi) \, \rho^2 \sin\varphi \, d\rho \, d\varphi \, d\theta[/latex]

The [latex]\rho^2[/latex] accounts for radial expansion (things get bigger farther from the origin), while [latex]\sin\varphi[/latex] accounts for how the circumference at angle [latex]\varphi[/latex] from the [latex]z[/latex]-axis changes.

When to Use Spherical Coordinates

Look for expressions like [latex]x^2 + y^2 + z^2[/latex], which becomes simply [latex]\rho^2[/latex] in spherical coordinates. Spherical coordinates are ideal for spheres, cones pointing along the [latex]z[/latex]-axis, and regions centered at the origin. A sphere [latex]x^2 + y^2 + z^2 = c^2[/latex] becomes [latex]\rho = c[/latex]. A cone [latex]z = \sqrt{x^2 + y^2}[/latex] becomes [latex]\varphi = \frac{\pi}{4}[/latex].

Setting Up Bounds

For a spherical box, ranges are straightforward: [latex]a \leq \rho \leq b[/latex], [latex]\alpha \leq \theta \leq \beta[/latex], and [latex]\gamma \leq \varphi \leq \psi[/latex]. For general regions, project onto the [latex]\rho\theta[/latex]-plane and determine how [latex]\varphi[/latex] varies. Find intersections by converting surface equations to spherical form and solving for the variables.

Problem-Solving Strategy

1. Convert surface equations to spherical coordinates
2. Identify geometric boundaries (spheres become constants for [latex]\rho[/latex], cones become constants for [latex]\varphi[/latex])
3. Find intersection points to determine ranges
4. Set up the integral with the volume element [latex]\rho^2 \sin\varphi , d\rho , d\varphi , d\theta[/latex]
5. Use symmetry when possible to simplify calculations