Volume of a solid via the divergence theorem

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Problem 4382 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.16 "The Ostrogradsky Formula"

Prove that the volume enclosed by the surface $S$ equals

V = \dfrac {1}{3}\iint _{S} (x\cos \alpha +y\cos \beta +z\cos \gamma )\, dS ,

where $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$ .

Problem 4383 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.16 "The Ostrogradsky Formula"

Prove that the volume of the cone bounded by the smooth conical surface $F(x,y,z)=0$ and the plane $Ax+By+Cz+D=0$ equals

V = \dfrac {1}{3}SH ,

where $S$ is the area of the base of the cone lying on the given plane, and $H$ is the height of the cone.

Problem 4384 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.16 "The Ostrogradsky Formula"

Find the volume of the solid bounded by the surfaces $z=\pm c$ and

x = a\cos u\cos v+b\sin u\sin v ,

y = a\cos u\sin v-b\sin u\cos v ,

z = c\sin u

Problem 4385 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.16 "The Ostrogradsky Formula"

Find the volume of the body bounded by the surface

x=u\cos v , \qquad y=u\sin v , \qquad z=-u+a\cos v \qquad (u\geqslant 0)

and the planes $x=0$ , $z=0$ $(a>0)$ .

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.