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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.

လုပ်ဆောင်ခဲ့သောဥပမာ: integrate x^2 dx from 0 to 1

Calculus · Definite-Integrals

Integrate x^2 from 0 to 1

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

ခြေလှမ်းတစ်လှမ်း

  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.

အဖြေကို ဖော်ပြပါ
\frac{1}{3}

Symbols used here

\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\nabla f
gradient (nabla, del)
Vector of partial derivatives; points uphill.
\iint,\ \oint
double / contour integral
Integral over a region of the plane; integral around a closed curve.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.
\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|
dot product, norm
Σ u_i v_i; the length of v, √(v·v).

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.

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ပိုပြီး Multivariable Calculus

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