Volume as a surface integral 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 SS equals

V=13∬S(xcos⁡α+ycos⁡β+zcos⁡γ) dS, V = \dfrac {1}{3}\iint _{S} (x\cos \alpha +y\cos \beta +z\cos \gamma )\, dS ,

where cos⁡α\cos \alpha, cos⁡β\cos \beta, cos⁡γ\cos \gamma are the direction cosines of the outward normal of the surface SS.

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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)=0F(x,y,z)=0 and the plane Ax+By+Cz+D=0Ax+By+Cz+D=0 equals

V=13SH, V = \dfrac {1}{3}SH ,

where SS is the area of the base of the cone lying on the given plane, and HH is the height of the cone.

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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=±cz=\pm c and

x=acos⁡ucos⁡v+bsin⁡usin⁡v, x = a\cos u\cos v+b\sin u\sin v , y=acos⁡usin⁡v−bsin⁡ucos⁡v, y = a\cos u\sin v-b\sin u\cos v , z=csin⁡u z = c\sin u

.

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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=ucos⁡v,y=usin⁡v,z=−u+acos⁡v(u⩾0) x=u\cos v , \qquad y=u\sin v , \qquad z=-u+a\cos v \qquad (u\geqslant 0)

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

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Problem 3769 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 9.11 "Integrals of Field Theory and Potential Theory"

Find the value of the integral ∬S(xcos⁡α+ycos⁡β+zcos⁡γ) ds\displaystyle \iint \limits_{S}(x\cos \alpha +y\cos \beta +z\cos \gamma ) \, ds, where α\alpha, β\beta, γ\gamma are the angles of the outward normal to the surface SS, and the integral is taken over the entire surface SS of some body of volume vv.

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Problem 3.11.49 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.11.2 "Problems"

Prove the formula for the volume VV of a body bounded by a smooth surface SS

V=∣13∬Sxdydz+ydzdx+zdxdy∣ V = \left|\frac{1}{3} \iint _{S} x d y d z+y d z d x+z d x d y\right|
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Problem 3.11.50 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.11.2 "Problems"

Using the formula from problem 49, find the volume of the body bounded by:

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(1)

the surface x=ucos⁡v,y=usin⁡v,z=−u+acos⁡v(u⩾0x = u \cos v, y = u \sin v, z = -u+a \cos v(u \geqslant 0, a>0)a > 0) and the planes x=0,z=0x = 0, z = 0;

(2)

the surface x=(b+acos⁡u)cos⁡v,y=(b+acos⁡u)sin⁡v;z=asin⁡u,b⩾a>0x = (b+a \cos u) \cos v, y = (b+a \cos u) \sin v ; z = a \sin u, b \geqslant a > 0

(3)

the surface x=acos⁡ucos⁡v+bsin⁡usin⁡v,y=acos⁡usin⁡v−−bsin⁡ucos⁡v,z=csin⁡ux = 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 and the planes z=c,z=−cz = c, z = -c.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.