Use cylindrical coordinates to find the volume of the solid using triple integrals

Mathematics Asked by Eric Brown on December 12, 2020

The given rectangular equations are
Converting to cylindrical coordinates I get
So my triple integral is
It’s a long and tedious calculation but I worked through it and got $-frac{1408pi}{15}$ which doesn’t make sense because you can’t have a negative volume. The negative came from the second integral which is
where I used a u-sub, where $u=16-r^2$ and $du=-2r$ and moving that $-frac{1}{2}$to the outside gives me
So could I just swap the limits of integration to get rid of it or have I messed up somewhere else?

2 Answers

There are two mistakes I notice. You have written wrong bound of $z$ and the equation of cylinder that was already pointed out. The correct integral should be

$V = 4displaystyle int_0^{frac{pi}{2}}int_0^{8cos(theta)}int_{0}^{sqrt{64-r^2}}rdzdrdtheta approx 617.22$

$V = 4displaystyle int_0^{frac{pi}{2}}int_0^{8cos(theta)} r , sqrt{64-r^2} , dr , dtheta$

On substition, $64-r^2 = u, rdr = -frac{1}{2}du$.

The bound $r = 0$ becomes $u = 64$, $r = 8 cos theta$ becomes $u = 64 sin ^2 theta$.

$V = -2displaystyle int_0^{frac{pi}{2}}int_{64}^{64sin^2(theta)} sqrt u , du , dtheta = 2int_0^{frac{pi}{2}}int_{64sin^2(theta)}^{64} sqrt u , du , dtheta$

$V = displaystyle frac{2048}{3} int_0^{frac{pi}{2}} (1-sin^3 theta) , dtheta = frac{2048}{3} (frac{pi}{2} - frac{2}{3})$

Correct answer by Math Lover on December 12, 2020

Continuing @MathLOver's calculation, the volume is$$begin{align}4int_0^{pi/2}dthetaint_0^{8costheta}rsqrt{64-r^2}dr&=4int_0^{pi/2}dthetaleft[-frac13(64-r^2)^{3/2}right]_0^{8costheta}\&=frac{2^{11}}{3}int_0^{pi/2}dtheta(1-sin^3theta)\&=frac{2^{11}}{3}int_0^{pi/2}dtheta(1-sintheta+sinthetacos^2theta)\&=frac{2^{11}}{3}left[theta+costheta-frac13cos^3thetaright]_0^{pi/2}\&=frac{1024}{9}left(3pi-4right)\&approx617.22.end{align}$$

Answered by J.G. on December 12, 2020

Add your own answers!

Related Questions

Most General Unifier computation

1  Asked on February 16, 2021 by milano


$sigma(n)$ is injective?

1  Asked on February 14, 2021 by ferphi


Find area of equilateral $Delta ABC $

2  Asked on February 14, 2021 by ellen-ellen


Diagonal of parallelogram and parallelepiped

0  Asked on February 13, 2021 by return


Ask a Question

Get help from others!

© 2022 All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir