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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · integrate x^2 dx from 0 to 1

Calculus · Definite-Integrals

Integrate x^2 from 0 to 1

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

Àwọn ìṣàmúlò-ètò

\int_{0}^{1} x^{2}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
\int x^{2}\, dx = \frac{x^{3}}{3}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
F(1) - F(0) = \left(\frac{1}{3}\right) - \left(0\right)
Fundamental theorem of calculus: plug in the limits.
= \frac{1}{3}
Simplify.

Fi àwọn àgbèwọlé hàn

\frac{1}{3}

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Bii o ṣe le: Double and triple integrals

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Multivariable Calculus

Functions of several variables Partial derivatives and the gradient Optimisation in several variables Vector fields, line integrals and the big theorems