Double Integral

The volume under a surface z = f(x, y) over a region in the plane: a rectangle, the region between two curves, or a polar region such as a disk or a cardioid. Optionally the volume between two surfaces. See the solid in 3D, the exact value step by step (inner integral, then outer integral) and the area of the region and the average height of f. Or switch to indefinite to find an antiderivative F(x, y) with ∂²F/∂x∂y = f.

Calculus Volume Multivariable

Inputs

Input examples

Type of integral

Between two surfaces

The volume enclosed by a top surface f(x, y) and a bottom surface h(x, y): ∬ (f − h) dA

f(x, y) — the height of the surface

Region R

x from

x to

y from

y to

What is a Double Integral?

A single integral adds up thin strips to give an area. A double integral adds up thin columns standing on a region R of the plane, each with height f(x, y) and a tiny base dA, so it gives the volume of the solid between the surface and the plane. Where f is negative that part counts as negative volume.

You compute it as an iterated integral (Fubini’s theorem): first integrate over the inner variable with the outer one held fixed, which gives the area of one slice, then integrate those slice areas over the outer variable. For a region between two curves the inner bounds are those curves, such as y from x² to x.

Round regions are easier in polar coordinates: x = r cos θ, y = r sin θ, and the small area becomes dA = r dr dθ. A disk of radius 2 is simply 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, and x² + y² becomes r².

f = 1gives the area of R

f ≥ 0gives a volume

÷ areaaverage height of f

polardA = r dr dθ

Where Double Integrals Are Used

Area between two curves. Integrating 1 over the region between a lower curve g(x) and an upper curve f(x) gives its area: ∬ 1 dA = ∫ (f(x) − g(x)) dx. Where the curves cross you usually cannot solve by hand (a parabola and a sine wave): the crossing finder does it numerically.

Volume between two surfaces. The solid between a top surface f and a bottom surface h, such as a dome cut off by a flat floor, has volume ∬ (f − h) dA. Tanks, lenses and earthworks are measured this way.

Averages and probability. The average of f over R is ∬ f dA divided by the area. In statistics the probability that (X, Y) falls in R is the double integral of their joint density over R.

The Solid in 3D

drag to turn

f(x, y)

h(x, y)

the volume that is calculated

faint: outside R (not counted)

Result

Enter a function and a region above

Step-by-Step Solution

The steps appear here after you calculate.

The Region R

xy-plane

Setting Up the Bounds

Between curves, dy dx

Draw a vertical line through R: it enters at the lower curve g₁(x) and leaves at the upper curve g₂(x). Those are the y bounds; the x bounds are the leftmost and rightmost x of R, for example where the curves meet.

Between curves, dx dy

Use a horizontal line instead: x runs from the left curve h₁(y) to the right curve h₂(y), and y over its full range. Some regions are much easier one way than the other.

Polar

For disks, rings and sectors: θ between two angles, r from an inner to an outer radius that may depend on θ (a cardioid has r from 0 to 1 + cos θ). Never forget the extra factor r in dA = r dr dθ.

Sources & Verification

References

J. Stewart, Calculus: Early Transcendentals, 8th ed., Cengage, 2016 — Ch. 15 (Multiple Integrals): double integrals over general regions, iterated integrals and polar coordinates.
M. Abramowitz & I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1964 — Ch. 25: Gauss–Legendre quadrature, used here for the numerical value.