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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.
कार्यरत उदाहरण · integrate x^2 dx from 0 to 1
चरणानुक्रमे
- \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.
उत्तर उघडा
आता तुम्ही प्रश्न निवडा, किंवा स्वतःचा प्रश्न टाइप करा किंवा रेखाटून दाखवा. प्रत्येक टप्प्यात, चित्रे आहेत, तुम्ही विचारण्याआधी उत्तर लपवलेले आहे.
मोफत खाते प्रत्येक शिकवणी वर टिपण्णी जोडते, तुम्ही पूर्ण केलेल्या गोष्टीचा नोंदवही, तुमच्या सोडविलेल्या समस्या एका ठिकाणी, आणि या पानाच्या विषयी तुम्ही विचारू शकता अशा शिक्षकाला. गणित स्वतःच सर्वांसाठी खुले आहे, दाखलन झालेले असो किंवा नसो.
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कसे कराल: 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.
लोक विचारतात ते प्रश्न
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.
अधिक माहिती Multivariable Calculus
Functions of several variablesPartial derivatives and the gradientOptimisation in several variablesVector fields, line integrals and the big theorems