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Deriving the volume of a right circular cylinder using triple integral
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Just for fun, I was wondering how one could derive the formula for the volume of a right circular cylinder using triple integrals in cylindrical coordinates.
calculus integration multivariable-calculus volume
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
asked Mar 13 '16 at 23:33
12332111233211
317111
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Just for fun, I was wondering how one could derive the formula for the volume of a right circular cylinder using triple integrals in cylindrical coordinates.
calculus integration multivariable-calculus volume
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
asked Mar 13 '16 at 23:33
12332111233211
317111
$endgroup$
add a comment |
$begingroup$
Just for fun, I was wondering how one could derive the formula for the volume of a right circular cylinder using triple integrals in cylindrical coordinates.
calculus integration multivariable-calculus volume
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
asked Mar 13 '16 at 23:33
12332111233211
317111
$endgroup$
Just for fun, I was wondering how one could derive the formula for the volume of a right circular cylinder using triple integrals in cylindrical coordinates.
calculus integration multivariable-calculus volume
calculus integration multivariable-calculus volume
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
asked Mar 13 '16 at 23:33
12332111233211
317111
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
asked Mar 13 '16 at 23:33
12332111233211
317111
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
edited Mar 13 '16 at 23:42
Ivo Terek
46.3k954142
46.3k954142
asked Mar 13 '16 at 23:33
12332111233211
317111
asked Mar 13 '16 at 23:33
12332111233211
317111
asked Mar 13 '16 at 23:33
12332111233211
317111
317111
add a comment |
add a comment |
2 Answers
2
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oldest
votes
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Let $r$ be the radius and $h$ be the height. The ranges for each variable are given by $rho in [0,r], z in [0,h], phi in ]-pi,pi]$. The volume van be evaluated using a triple integral
$$begin{align*}
iiint_{mathrm{Cyl}} dV
&= int_0^r int_0^h int_{-pi}^pi rho ,dphi,dz,drho \
&= int_0^r int_0^h 2pi rho,dz,drho \
&= int_0^r 2pi rho h ,drho \
&= left. pi rho^2 h right|_0^r = pi r^2 h
end{align*}$$
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
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add a comment |
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Let $C = { (x,y,z) in Bbb R^3 mid x^2+y^2 leq R^2, ~0leq z leq h }$. If you put $f(r, theta, z) = (rcostheta, rsintheta, z)$, then by the change of variables theorem we'll have: $$begin{align} {rm Vol}(C) &= int_C 1 = int_{f([0,R]times[0,2pi]times[0,h])}1 \ &= int_{[0,R]times[0,2pi]times[0,h]}|det Df(r,theta,z)|,{rm d}r,{rm d} theta,{rm d}z \ &= int_0^hint_0^{2pi}int_0^R r ,{rm d}r,{rm d}theta,{rm d}z \ &=2pi h frac{r^2}{2}bigg|_0^R = pi R^2h. end{align}$$
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
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2 Answers
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active
oldest
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2 Answers
2
active
oldest
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active
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active
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votes
$begingroup$
Let $r$ be the radius and $h$ be the height. The ranges for each variable are given by $rho in [0,r], z in [0,h], phi in ]-pi,pi]$. The volume van be evaluated using a triple integral
$$begin{align*}
iiint_{mathrm{Cyl}} dV
&= int_0^r int_0^h int_{-pi}^pi rho ,dphi,dz,drho \
&= int_0^r int_0^h 2pi rho,dz,drho \
&= int_0^r 2pi rho h ,drho \
&= left. pi rho^2 h right|_0^r = pi r^2 h
end{align*}$$
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
$endgroup$
add a comment |
$begingroup$
Let $r$ be the radius and $h$ be the height. The ranges for each variable are given by $rho in [0,r], z in [0,h], phi in ]-pi,pi]$. The volume van be evaluated using a triple integral
$$begin{align*}
iiint_{mathrm{Cyl}} dV
&= int_0^r int_0^h int_{-pi}^pi rho ,dphi,dz,drho \
&= int_0^r int_0^h 2pi rho,dz,drho \
&= int_0^r 2pi rho h ,drho \
&= left. pi rho^2 h right|_0^r = pi r^2 h
end{align*}$$
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
$endgroup$
add a comment |
$begingroup$
Let $r$ be the radius and $h$ be the height. The ranges for each variable are given by $rho in [0,r], z in [0,h], phi in ]-pi,pi]$. The volume van be evaluated using a triple integral
$$begin{align*}
iiint_{mathrm{Cyl}} dV
&= int_0^r int_0^h int_{-pi}^pi rho ,dphi,dz,drho \
&= int_0^r int_0^h 2pi rho,dz,drho \
&= int_0^r 2pi rho h ,drho \
&= left. pi rho^2 h right|_0^r = pi r^2 h
end{align*}$$
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
$endgroup$
Let $r$ be the radius and $h$ be the height. The ranges for each variable are given by $rho in [0,r], z in [0,h], phi in ]-pi,pi]$. The volume van be evaluated using a triple integral
$$begin{align*}
iiint_{mathrm{Cyl}} dV
&= int_0^r int_0^h int_{-pi}^pi rho ,dphi,dz,drho \
&= int_0^r int_0^h 2pi rho,dz,drho \
&= int_0^r 2pi rho h ,drho \
&= left. pi rho^2 h right|_0^r = pi r^2 h
end{align*}$$
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
answered Mar 13 '16 at 23:38
Henricus V.Henricus V.
15.2k22149
15.2k22149
add a comment |
add a comment |
$begingroup$
Let $C = { (x,y,z) in Bbb R^3 mid x^2+y^2 leq R^2, ~0leq z leq h }$. If you put $f(r, theta, z) = (rcostheta, rsintheta, z)$, then by the change of variables theorem we'll have: $$begin{align} {rm Vol}(C) &= int_C 1 = int_{f([0,R]times[0,2pi]times[0,h])}1 \ &= int_{[0,R]times[0,2pi]times[0,h]}|det Df(r,theta,z)|,{rm d}r,{rm d} theta,{rm d}z \ &= int_0^hint_0^{2pi}int_0^R r ,{rm d}r,{rm d}theta,{rm d}z \ &=2pi h frac{r^2}{2}bigg|_0^R = pi R^2h. end{align}$$
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
$endgroup$
add a comment |
$begingroup$
Let $C = { (x,y,z) in Bbb R^3 mid x^2+y^2 leq R^2, ~0leq z leq h }$. If you put $f(r, theta, z) = (rcostheta, rsintheta, z)$, then by the change of variables theorem we'll have: $$begin{align} {rm Vol}(C) &= int_C 1 = int_{f([0,R]times[0,2pi]times[0,h])}1 \ &= int_{[0,R]times[0,2pi]times[0,h]}|det Df(r,theta,z)|,{rm d}r,{rm d} theta,{rm d}z \ &= int_0^hint_0^{2pi}int_0^R r ,{rm d}r,{rm d}theta,{rm d}z \ &=2pi h frac{r^2}{2}bigg|_0^R = pi R^2h. end{align}$$
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
$endgroup$
add a comment |
$begingroup$
Let $C = { (x,y,z) in Bbb R^3 mid x^2+y^2 leq R^2, ~0leq z leq h }$. If you put $f(r, theta, z) = (rcostheta, rsintheta, z)$, then by the change of variables theorem we'll have: $$begin{align} {rm Vol}(C) &= int_C 1 = int_{f([0,R]times[0,2pi]times[0,h])}1 \ &= int_{[0,R]times[0,2pi]times[0,h]}|det Df(r,theta,z)|,{rm d}r,{rm d} theta,{rm d}z \ &= int_0^hint_0^{2pi}int_0^R r ,{rm d}r,{rm d}theta,{rm d}z \ &=2pi h frac{r^2}{2}bigg|_0^R = pi R^2h. end{align}$$
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
$endgroup$
Let $C = { (x,y,z) in Bbb R^3 mid x^2+y^2 leq R^2, ~0leq z leq h }$. If you put $f(r, theta, z) = (rcostheta, rsintheta, z)$, then by the change of variables theorem we'll have: $$begin{align} {rm Vol}(C) &= int_C 1 = int_{f([0,R]times[0,2pi]times[0,h])}1 \ &= int_{[0,R]times[0,2pi]times[0,h]}|det Df(r,theta,z)|,{rm d}r,{rm d} theta,{rm d}z \ &= int_0^hint_0^{2pi}int_0^R r ,{rm d}r,{rm d}theta,{rm d}z \ &=2pi h frac{r^2}{2}bigg|_0^R = pi R^2h. end{align}$$
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
answered Mar 13 '16 at 23:41
Ivo TerekIvo Terek
46.3k954142
46.3k954142
add a comment |
add a comment |
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