IGNOU BMTC 102 2026 - Use double integration of find the volume of the ellipsoid

Question

Use double integration of find the volume of the ellipsoid

$$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1.$

09 Jan 2026

Answer :

Word Count : 287

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} - \frac{y^2}{9}}. ] The volume is [ V = \iint_R 2 z , dx , dy = \iint_R 8 \sqrt{1 - \frac{x^2}{4} - \frac{y^2}{9}} , dx , dy, ] where (R) is the projection on the (xy)-plane: [ \frac{x^2}{4} + \frac{y^2}{9} \le 1. ] Use the substitution (x = 2 u), (y = 3 v), so that (dx,dy = 6 , du,dv) and the region becomes

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