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Double and triple integrals

Volume under a surface by iterated integration; changing the order and the coordinates.

∫∫ f dA sums f over a region by integrating one variable at a time (Fubini). Polar coordinates turn circles into rectangles and give the famous ∫e^{−x²} = √π. Picture it: the volume under the surface z = f(x, y) over the region. Think it: the Jacobian is the local scaling factor (the determinant of the derivative), which is why r appears in polar integrals.

Worked example · integrate x^2 dx from 0 to 1

Calculus · Definite-Integrals

Integrate x^2 from 0 to 1

\int_{0}^{1} x^{2}\, dx

Step by step

  1. \int_{0}^{1} x^{2}\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int x^{2}\, dx = \frac{x^{3}}{3}

    Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

  3. F(1) - F(0) = \left(\frac{1}{3}\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = \frac{1}{3}

    Simplify.

Reveal the answer
\frac{1}{3}

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How to: Double and triple integrals

  1. First find an antiderivative F, then evaluate F(b) − F(a).
  2. Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
  3. Fundamental theorem of calculus: plug in the limits.
  4. Simplify.

Questions people ask

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen: the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

More in Multivariable Calculus

Functions of several variablesPartial derivatives and the gradientOptimisation in several variablesVector fields, line integrals and the big theorems