MTH2010 Chap.8 Triple Integrals in Cylindrical and Spherical Coordinates
Triple Integrals in Cylindrical and Spherical Coordinates
Week 8 moves from regions in the plane to solids in space. A triple integral is again the limit of Riemann sums, now over small boxes, and integrating 1 gives the volume of the solid. To evaluate one in Cartesian form, the slides classify solids by which variable runs between two surfaces.
A type I solid has z between a lower surface u(x, y) and an upper surface v(x, y) above a region D in the plane; types II and III do the same for x and for y. The inner integral reduces the problem to a double integral over D, and a three-dimensional form of Fubini's theorem allows different orders whenever the solid has more than one type. Two curvilinear systems make symmetric solids manageable.
Cylindrical coordinates keep z and use polar coordinates in the plane, so the volume element is r dr dθ dz and solids with an axis of symmetry have simple limits. Spherical coordinates use the distance ρ from the origin, the angle φ measured down from the positive z-axis and the azimuthal angle theta, so spheres and cones about the z-axis become coordinate surfaces and the volume element is ρ² sin(φ) dρ dθ dφ.
One homework solution swaps the names of the two angles, so stating a convention at the start of each answer is worthwhile. The applied sheet for this material also defines the average value of a function over a solid and reduces solids of revolution to a single integral.
What this chapter covers
- 01
Triple integrals as limits of sums over small boxes
- 02
Region types I, II and III and the inner integral
- 03
Volume as the integral of one, and average value
- 04
Cylindrical coordinates: dV = r dr dθ dz
- 05
Spherical coordinates: rho, φ and θ and their ranges
- 06
Spherical volume element: ρ² sin(φ)
- 07
Choosing the system from the shape of the solid
- 08
Solids of revolution reduced to one integral
Worked example · free
A spherical integral over the unit ball
- 1In spherical form z = ρ cos(φ) and dV = ρ² sin(φ) dρ dφ dθ, with 0 ≤ ρ ≤ 1, 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2 π.
- 1The integrand becomes rho⁴ cos^2(φ) sin(φ), which separates into three factors.
- 1Factors: the rho-integral is 1/5, the phi-integral of cos² sin is 2/3, and the theta-integral is 2 π.
- 1Answer: (1/5)(2/3)(2π) = 4 π/15.
Key terms
- Triple integral
- The limit of sums of function values times small box volumes over a solid region.
- Cylindrical coordinates
- Coordinates that describe a point by its polar coordinates in the plane together with its height z.
- Spherical coordinates
- Coordinates that describe a point by its distance from the origin and two angles.
- Average value
- The integral of a function over a solid divided by the volume of that solid.
Triple Integrals in Cylindrical and Spherical Coordinates FAQ
How do I choose between cylindrical and spherical coordinates?
Use cylindrical coordinates for solids with an axis of symmetry, such as cylinders, paraboloids and cones described by z and r. Use spherical coordinates when spheres centred at the origin or cones about the z-axis bound the solid.
Which angle is φ in spherical coordinates?
The seminar slides measure φ down from the positive z-axis and use θ for the angle in the xy-plane. Some sources swap the names, so state your convention before writing limits.
What is a type I solid?
A solid lying between a lower surface and an upper surface over a region in the xy-plane. Integrate z first between the two surfaces, then integrate the result over the region.
How can I check a volume answer?
Compare it with simple solids that sit inside or around it, such as a cylinder or a cone, or recompute it in another coordinate system when the solid allows.
Exam move
For each solid, sketch a cross-section before choosing coordinates; the cross-section shows which variable runs between two surfaces and where the angles start and stop. Practise one cylindrical problem where the height depends on r and one spherical problem with a cone boundary, and always write the volume element on its own line. After each answer, check plausibility against a simple enclosing solid.
Finally, rehearse the solid-of-revolution reduction, which links the three-dimensional set-up back to a familiar single integral.
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