What is the triple integral?
Type f(x, y, z), the order such as dz dy dx, and the limits from the inside out. Inner limits may use the outer letters, so solids such as tetrahedra work.
- Exact value
- 1/12
The triple integral of 5x - 3y (dz dy dx) is 0.08333333333.
- Decimal value
- 0.08333333333
Exact value: 1/12. The triple integral of 5x - 3y (dz dy dx) is 0.08333333333.
How to calculate
Evaluates an iterated integral exactly where possible, checked numerically.
Example with the default inputs (Function f(x, y, z) 5x - 3y, Order dz dy dx, Inner lower 0, Inner upper 1 - x - y, Middle lower 0, Middle upper 1 - x, Outer lower 0, Outer upper 1): The triple integral of 5x - 3y (dz dy dx) is 0.08333333333.
Method: Iterated CAS definite integrals, checked against tanh-sinh cubature.
- Finite limits; radians. An answer that fails its check is not shown.
Worked examples
Each example is checked against the calculator on every build.
- Function f(x, y, z) 5x - 3y, Order dz dy dx, Inner lower 0, Inner upper 1 - x - y, Middle lower 0, Middle upper 1 - x, Outer lower 0, Outer upper 1 gives Exact value 1/12, Decimal value 0.083333.Source: OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.4, Ex. 5.38
- Function f(x, y, z) 1, Order dx dy dz, Inner lower -1 + z, Inner upper 1 - z, Middle lower -1 + z, Middle upper 1 - z, Outer lower 0, Outer upper 1 gives Exact value 4/3, Decimal value 1.333333.Source: OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.4, Ex. 5.39
How it works
The triple integral calculator evaluates the iterated integral of f over a solid, in the order you type. With dz dy dx:
∫ from a₃ to b₃ ( ∫ from a₂(x) to b₂(x) ( ∫ from a₁(x, y) to b₁(x, y) f(x, y, z) dz ) dy ) dx.
Other orders work the same way: the first letter of the order is the inner one. The inner limits may use the middle and outer letters, the middle limits may use the outer letter, and the outer limits are numbers or constant expressions (pi, e, sqrt(2)). A lower limit above the upper limit changes the sign, as for one variable. All limits must be finite.
Exact value. A computer algebra system (nerdamer, open source) works out each integral in turn, from the inside out, as a checked definite integral in its letter with the outer letters held constant; each result is the integrand of the next integral. The algebra runs after you start typing, in the background. When every sin and cos of the letter u being integrated has the form k·u + b, where u appears once, as a factor of the one term k·u, and k and b are free of u, that letter is first replaced by w = k·u, and sin(w + b), cos(w + b) are split by the angle-sum rules; the value does not change.
Check. The page also integrates f numerically: a tanh-sinh rule in each letter, with nodes at t = −3 to 3 in steps of 1/8 (49 per letter), each inner rule over its limits at the outer nodes. The exact value must agree with this to 10⁻⁸ × max(1, |value|).
When there is no exact value. If an inner integral has no exact form found, the exact answer fails its check, or the algebra takes over 3 seconds, the page shows the numeric value alone (Decimal value, no Exact value), but only when the rule above and the same rule with steps of 1/12 agree to 10⁻¹⁰ × max(1, |value|); otherwise "No verified answer". When the algebra finds that an integral does not exist (a pole inside an interval, an end where the antiderivative is infinite, or a function that is not real on part of the region), the page says so and shows no value.
What you can type
- The order: d and a letter for each of the three variables, inner first, such as dz dy dx or dx dy dz. Letters can be any lowercase letters except e (Euler’s number), and θ.
- The function may use only the three letters of the order. pi (or π) is π and e is Euler’s number.
- Numbers can have decimals (2.5) and powers of ten (1e-3). Operations: + − * / and ^; brackets group; numbers and letters side by side multiply (2x y z).
- Functions: sqrt, cbrt, ln (and log, the natural logarithm), log10, exp, abs, sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh and their inverses. Angles are in radians.
How answers are written
- Exact value: the algebra’s exact form, in the syntax you type, shown as MathML. It is left out when its text holds a decimal point or a whole number of 9 or more digits (a number the algebra rounded). It is not simplified further.
- Decimal value: the value to 10 significant figures.
Assumptions
- Angles are in radians; ln and log are the natural logarithm.
- The solid is the one the limits describe.
- An answer that fails its check is not shown ("No verified answer").
Worked examples by hand
∭ (5x − 3y) dV over the tetrahedron 0 ≤ z ≤ 1 − x − y, 0 ≤ y ≤ 1 − x, 0 ≤ x ≤ 1 (OpenStax Calculus Volume 3, section 5.4, Example 5.38). Inner: (5x − 3y)(1 − x − y). Middle, y from 0 to 1 − x: with u = 1 − x, ∫ (5x − 3y)(u − y) dy from 0 to u = 5xu²/2 − u³/2. Outer, x from 0 to 1: ∫ (5x(1 − x)² − (1 − x)³)/2 dx = (5/12 − 1/4)/2 = 1/12.
∭ 1 dV over 0 ≤ z ≤ 1, −1 + z ≤ y ≤ 1 − z, −1 + z ≤ x ≤ 1 − z (Example 5.39), order dx dy dz. Inner: 2(1 − z). Middle: 4(1 − z)². Outer: ∫ 4(1 − z)² dz from 0 to 1 = 4/3, the volume of a pyramid with a 2 × 2 base and height 1.
Other questions people ask
What is a triple integral?
A triple integral adds up f(x, y, z) over a solid region in space. With f = 1 it is the volume of the solid; with f a density it is the mass. It is worked out as an iterated integral: integrate in the inner letter with the other two held constant, then in the middle letter, then in the outer letter.
How do I enter the limits?
Type the order inner first, such as dz dy dx. The inner limits may use the two outer letters (here x and y), the middle limits may use the outer letter (x), and the outer limits must be numbers or constants. For the tetrahedron x, y, z ≥ 0, x + y + z ≤ 1, use z from 0 to 1 − x − y, y from 0 to 1 − x, and x from 0 to 1.
How is the answer checked?
The exact value comes from a computer algebra system, one integral at a time. The page also works out the triple integral numerically, with a tanh-sinh rule in each letter, and the two must agree to 1 part in 100 million. If no exact form is found, or it fails the check, the page shows the numeric value alone, and only when two runs of the numeric rule with different step sizes agree.
Why do I see only a decimal?
The algebra found no exact antiderivative for one of the integrals, its answer failed the check, or it took over 3 seconds. The decimal is the numeric value to 10 significant figures, shown only when two numeric runs agree to 1 part in 10 billion. Changing the order of integration sometimes gives an exact answer.
Can I use cylindrical or spherical coordinates?
Yes: use your own letters in the order and put the Jacobian factor into the function. In cylindrical coordinates multiply by r (order dz dr dθ); in spherical coordinates multiply by ρ² sin φ, typed with plain letters such as p^2 sin(q). The volume of a ball of radius 1 is p^2 sin(q) with dp dq dt, p from 0 to 1, q from 0 to pi and t from 0 to 2pi, which is 4π/3.
What is the difference from a double integral?
A double integral runs over a region of the plane and has two limits pairs; a triple integral runs over a solid and has three. For one-variable definite integrals use the integral calculator.