Integration in Polar, Cylindrical and Spherical Coordinates: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 7, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A plane coordinate system locating a point by its distance from the origin and the angle its ray makes with the positive x axis.
Polar to rectangular conversion
The x coordinate is the radius times the cosine of the angle and the y coordinate is the radius times the sine, which is the substitution used inside any polar integral.
Non-uniqueness of polar representation
The same point has infinitely many polar descriptions because the angle may shift by full turns and the radius may be negated with a half-turn added.
Polar area element
The small area in polar coordinates is the radius times the change in radius times the change in angle. Forgetting the extra radius factor is the single most common polar error.
Polar double integral setup
Angle limits describe the sweep of the region and radius limits describe entry and exit along each ray, with the integrand rewritten and the radius factor inserted.
Polar rectangle
A region bounded by two circular arcs and two rays, the polar analogue of a coordinate rectangle and the natural cell for polar Riemann sums.
When to choose polar coordinates
Whenever the region is a disk, annulus or circular sector, or the integrand contains the sum of the squares of the two variables, which collapses to a single squared radius.
Area between polar curves
Computed by integrating half the difference of the squared outer and inner radii over the angle interval where the outer curve genuinely lies outside.
Cylindrical coordinates
Polar coordinates in the horizontal plane paired with the unchanged height coordinate, ideal for solids with an axis of rotational symmetry.
Cylindrical volume element
The small volume is the radius times the change in radius, angle and height, inheriting the same extra radius factor as the polar area element.
Cylindrical description of a cone
A cone with vertex at the origin becomes a simple proportionality between height and radius, which is why cylindrical setups tame otherwise awkward cone problems.
Spherical coordinates
A system using distance from the origin, the polar angle measured down from the positive vertical axis, and the azimuthal angle measured in the horizontal plane.
Spherical polar angle range
The angle from the positive vertical axis runs from zero to a straight angle only; allowing a larger range would double count every point of space.
Spherical azimuthal angle range
The horizontal angle sweeps a full turn, exactly as in polar coordinates, to cover every direction around the vertical axis.
Spherical volume element
The small volume equals the squared radius times the sine of the polar angle times the changes in radius and both angles. Dropping the sine factor is a frequent and fatal slip.
Sphere in spherical coordinates
A sphere centred at the origin is the single equation setting the radius equal to a constant, which reduces its volume integral to constant limits throughout.
Cone in spherical coordinates
A cone with vertex at the origin and axis along the vertical becomes a constant polar angle, so ice cream cone solids get constant limits in two of three variables.
Choosing between cylindrical and spherical
Cylindrical suits solids with a symmetry axis and flat horizontal caps, while spherical suits solids bounded by spheres and cones sharing a common centre.
Order of variables in spherical integration
Radius innermost with the two angles outside is usually simplest, since entering and exiting surfaces are most often described as radius bounds.
Jacobian for polar coordinates
The change of variables determinant equals the radius, which is precisely where the extra radius factor in the polar area element comes from.
Jacobian for spherical coordinates
The determinant equals the squared radius times the sine of the polar angle, the source of the spherical volume element and a nonnegative quantity on the standard ranges.
Degenerate limits at the origin
The angle is undefined at the origin and the spherical element vanishes on the vertical axis, harmless for integration since these sets contribute no volume.
Symmetry shortcuts in curved coordinates
When both solid and integrand are symmetric about the vertical axis, the azimuthal integral contributes only a factor of a full turn and can be done immediately.
Improper multiple integral
An integral over an unbounded region or of an unbounded integrand, defined as a limit of integrals over expanding bounded pieces and convergent only if that limit is finite.
Gaussian integral by polar conversion
The classical trick of squaring a one-dimensional exponential integral, reading the product as a double integral over the plane, and evaluating it in polar coordinates.