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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.
مثال · integrate x^2 dx from 0 to 1
قدم ب قدم
- \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.
جواب کھوليں
اب آپ مسئلہ منتخب کريں يا ٹائپ کريں يا خود کريں ہر قدم ، تصوير ، جواب پوشيدہ هے جب تک آپ سوال نہ کريں
علامتیں
تمام تعريف کے ليے ، تصوير کے ليے اور اس ميں ہر حروف کا مطلب کے ليے کو ئي علامت کو ٹک کريں
کیسے: 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.
سوالات لوگ پوچھتے ہیں
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.
میں زیادہ Multivariable Calculus
Functions of several variablesPartial derivatives and the gradientOptimisation in several variablesVector fields, line integrals and the big theorems