IGNOU BMTC 102 2026 - Evaluate , where S is the solid region between the sphe

Question

Evaluate $\displaystyle \iiint_S z^2 \, dx \, dy \, dz$ , where S is the solid region between the spheres $\rho = 1$ and
$\quad$ $\rho = 2$ , by using spherical coordinates.

09 Jan 2026

Answer :

Word Count : 294

We are asked to evaluate [ \iiint_S z^2 , dx,dy,dz ] where (S) is the solid region between the spheres (\rho = 1) and (\rho = 2). We will solve this using spherical coordinates. --- In spherical coordinates: [ x = \rho \sin\phi \cos\theta, \quad y = \rho \sin\phi \sin\theta, \quad z = \rho \cos\phi ] and the volume element is: [ dx,dy,dz = \rho^2 \sin\phi , d\rho, d\phi, d\theta ] Also, (z^2 = (\rho \cos\phi)^2 = \rho^2 \cos^2\phi). Therefore, the integrand becomes: [ z^2 , dx,dy,dz = \rho^2 \cos^2\phi \cdot \rho^2 \sin\phi , d\rho, d\phi, d\theta = \rho^4 \cos^2\phi

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