Multivariable calculus

Triple integrals

Integrating over a solid region in three variables.

• Triple Integral of xyz over the Unit CubeThe triple integral of xyz over the unit cube is 1/8. See the iterated method, the factoring shortcut, and when that shortcut is not allowed.
• Triple Integral of x + y + z over a BoxThe triple integral of x + y + z over the box 0 to 1 by 0 to 2 by 0 to 3 is 18. Split the sum by linearity, then check it against the centre point.
• Triple Integral of x^2 + y^2 + z^2 over a BoxThe triple integral of x^2 + y^2 + z^2 over the box 0 to 1 by 0 to 2 by 0 to 1 is 4. This is the squared distance integral behind moments of inertia.
• Triple Integral of e^(x+y+z) over the Unit CubeThe triple integral of e^(x+y+z) over the unit cube is (e - 1)^3, about 5.073214. The exponent splits into a product of three identical integrals.
• Triple Integral of z sin x cos y over a BoxThe triple integral of z sin(x) cos(y) over 0 to pi/2 in x and y and 0 to 2 in z equals 2. A clean separable example with trigonometric limits.
• Triple Integral of x y^2 z^3 over a BoxThe triple integral of x y^2 z^3 over the box 0 to 1 by 0 to 2 by 0 to 1 is 1/3. The power rule in each variable, and the exponent slip to avoid.
• Triple Integral of xy + z^2 over a BoxThe triple integral of xy + z^2 over the box 0 to 1 by 0 to 2 by 0 to 3 is 21. A sum is not a product, so split it into two separable pieces first.
• Triple Integral of x/(1 + y^2) over a BoxThe triple integral of x/(1 + y^2) over the box 0 to 2 by 0 to 1 by 0 to 3 equals 3pi/2. The y factor is an arctangent, not a logarithm.
• Triple Integral of y ln(1 + x) over a BoxThe triple integral of y ln(1 + x) over the box 0 to 1 by 0 to 2 by 0 to 3 equals 12 ln 2 minus 6, about 2.317766. Integration by parts inside a box.
• Triple Integral of xyz + x^2 over a CubeOver the cube from -1 to 1 in every variable, the triple integral of xyz + x^2 is 8/3. The odd xyz term cancels and only the even term survives.