maths.free › Multivariable Calculus › Double and triple integrals
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.
Arbejdstxrd eksempel · integrate x^2 dx from 0 to 1
Trin for trin
- \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.
Afslør svaret
Nu er du Vælg et problem, eller skriv eller tegne din egen. Hvert skridt, et billede, svaret skjult, indtil du spørger.
En gratis konto tilføjer noter om hver lektion, en rekord over, hvad du har afsluttet, dine løste problemer på ét sted, og en vejleder, du kan spørge om denne side. Matematik selv er åben for alle, logget på eller ej.
Tilmeld dig LoginSymboler brugt her
Tryk på ethvert symbol for den fulde definition, et billede, og hvad hvert bogstav i det betyder.
Hvordan: 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.
Spørgsmål folk spørger
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.
Mere i Multivariable Calculus
Functions of several variablesPartial derivatives and the gradientOptimisation in several variablesVector fields, line integrals and the big theorems