# Find volume under given contraints on the Cartesian plane.

The constrains are given as

$$x^2+y^2+z^2 leqslant 64,,x^2+y^2leqslant 16,,x^2+y^2leqslant z^2,,zgeqslant 0,.$$

With the goal of finding the Volume.
Personally, I have trouble interpreting the constrains in terms of integrals to find the Volume. However, logically z can be maximum of 8
and x or y no larger than 4.

Mathematics Asked by Pavel Fedotov on November 12, 2021

The problem is simpler if you convert to cylindrical coordinates. In this coordinate system, we have $$rho=sqrt{x^2+y^2}$$, $$z=z$$, and $$phi=textrm{arctan}left(frac{y}{x}right)$$ (sort of - see the wikipedia article for a better explanation of how $$phi$$ relates to $$x$$ and $$y$$). An important point is that $$rho$$ is taken to be positive, while $$phi$$ dictates direction completely.

Under this system, we have the bounds $$rho^2+z^2leq 64, rho^2leq 16, rho^2leq z^2, zgeq 0$$. From these inequalities, we have that $$z$$ may range anywhere from $$0$$ to $$8$$, and there is no restriction on $$phi$$ (so it ranges from $$0$$ to $$2pi$$). Combining the inequalities involving $$rho$$, we get that as $$z$$ ranges from $$0$$ to $$4$$, $$rho$$ ranges from $$0$$ to $$z$$. Then, as $$z$$ ranges from $$4$$ to $$sqrt{48}$$, $$rho$$ ranges from $$0$$ to $$4$$. Finally, as $$z$$ ranges from $$sqrt{48}$$ to $$8$$, $$rho$$ ranges from $$0$$ to $$sqrt{64-z^2}$$.

We can write each of these three volumes as triple integrals. The volume element in cylindrical coordinates, $$dV$$, is $$rho;drho;dz;dphi$$. So the first will be

$$int_0^{2pi}int_0^4int_0^z 1times ; rho;drho;dz;dphi$$$Similarly the second will be $$int_0^{2pi}int_4^{sqrt{48}}int_0^4 1times ; rho;drho;dz;dphi$$ and the third will be $$int_0^{2pi}int_{sqrt{48}}^8int_0^{sqrt{64-z^2}} 1times ; rho;drho;dz;dphi$$ Solve each triple integral from the inside out, add the three together, and you'll have your answer. (feel free to comment or edit for any correction or suggestion) Answered by Cardioid_Ass_22 on November 12, 2021 Split the volume into three parts: cone, cylinder, and spherical cap: The formulas for these three-dimensional figures are well known: $$V_{cone} = frac{1}{3} pi r^2 h$$ $$V_{cylinder} = pi r^2 h$$ $$V_{cap} = frac{1}{3} pi a^2 (3 R - a)$$ where $$R$$ is the radius of the sphere and $$a$$ is the distance between the plane "cutting" the cap and the center of the sphere. All these variables are found easily through simple high-school algebra. Answered by David G. Stork on November 12, 2021 ## Add your own answers! ## Related Questions ### Does$prod_k (x – r_k)^{m_k} = prod_k (1 – frac{x}{r_k})^{m_k}$? 1 Asked on January 23, 2021 by novice ###$pi_2(T vee mathbb{C}P^2)$and action of$pi_1$on$pi_2$1 Asked on January 23, 2021 by urbanog ### Solving$a=x^p – (x-b)^p$for arbitrary$a,b$, and$p$being natural/rational? 0 Asked on January 23, 2021 by kristof-spenko ### Sylow’s second theorem explanation 1 Asked on January 23, 2021 by pritam ### Mathematical terminology 0 Asked on January 23, 2021 by jeremys ### Finding saddle point from the critical points 1 Asked on January 22, 2021 by lawrence-mano ###$1cdot3cdot5cdots(2n-1) < (frac{2n}{e})^{n+1}, n in mathbb{N}, n geq 2$proof doesn't seem to work 1 Asked on January 22, 2021 by testcase12 ### Ambiguity in the evaluation of a real integral multiplied by a complex number 1 Asked on January 22, 2021 by gabriele-privitera ### Construction of a connection on a vector bundle$Eto M$with trivial determinant bundle 1 Asked on January 22, 2021 ### Proving$tan^{-1}frac{1}{x} leq frac{1}{x}$3 Asked on January 22, 2021 by reece-mcmillin ### How can we show that there is a zero of$f$2 Asked on January 22, 2021 ### Convergence in vector space 1 Asked on January 22, 2021 ### Conditon for$f(x)$such that$f(d(x, y))$induces the same topology with$d(x, y)$1 Asked on January 22, 2021 by random487510 ### Simple$R\$-module equivalent statement

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