Use double integration of find the volume of the ellipsoid
The given ellipsoid is [ \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1. ] We want the volume using double integration. Solve for (z) in terms of (x) and (y): [ z^2 = 16 \left(1 - \frac{x^2}{4} - \frac{y^2}{9}\right) \implies z = \pm 4 \sqrt{1 - \frac{x^2}{4} - ___ ____ _____ _______ ____.
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Write as an integral over a region D. Sketch the region D and show that it is of both types 1 and 2. Reverse the order of integration and evaluate it.
Let be three points in
.
Find |2b - a + 3c|.
Evaluate , where S is the solid region between the spheres
and
, by using spherical coordinates.
ग्रीन प्रमेय का कथन दीजिए और इसकी सहायता से
जहाँ C, दीर्घवृत्त है।
(ख) पृष्ठ पर फलन
के चरम मान ज्ञात कीजिए।
Using only the definitions, find fxy(0, 0) and fyx(0, 0), if they exists, for the function
Find the following limits:(i)
(ii)
State whether the following statements are true or false. Give reasons for your answers.
(i)
(ii) A real-valued function of three variables which is continuous everywhere is differentiable.
(iii) The function , defined by
, is locally invertible at any
.
(iv) , defined by
is integrable.
(v) The function , defined by
, has an extremum at (0, 0).