Integration over portion of circle
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I need to find the integral region $mathcal R$ which covers area CPRQ portion of the circle radius $t$.
The integration looks as
$$I=intint_mathcal Rf(r,theta)dr dtheta$$
Can someone help me to write these polar coordinate integration limits for $r$ and $theta$?
If necessary, he angle $RCQ$ is $beta$ where $Tan, beta=fracdh$.
The angle $ROQ$ is $alpha$ where
$cos,alpha = frach left(sqrtd^2+h^2 sqrtt^2-fracd^2 h^2d^2+h^2+d^2right)t left(d^2+h^2right)$
linear-algebra integration circle polar-coordinates
asked Aug 20 at 5:25
Frey
616312
add a comment |Â
up vote
0
down vote
favorite
I need to find the integral region $mathcal R$ which covers area CPRQ portion of the circle radius $t$.
The integration looks as
$$I=intint_mathcal Rf(r,theta)dr dtheta$$
Can someone help me to write these polar coordinate integration limits for $r$ and $theta$?
If necessary, he angle $RCQ$ is $beta$ where $Tan, beta=fracdh$.
The angle $ROQ$ is $alpha$ where
$cos,alpha = frach left(sqrtd^2+h^2 sqrtt^2-fracd^2 h^2d^2+h^2+d^2right)t left(d^2+h^2right)$
linear-algebra integration circle polar-coordinates
asked Aug 20 at 5:25
Frey
616312
Are you familiar with the equation for a line not through the origin in polar coordinates?
– David K
Aug 20 at 5:49
@David I am not much familiar but I have looked at some examples such as math.stackexchange.com/questions/2430448/… I am not sure how it can be applied in this case.
– Frey
Aug 21 at 7:13
Sadly, that question-asker was confused; the question did not actually involve polar coordinates. Try math.stackexchange.com/questions/1382437/… instead.
– David K
Aug 21 at 11:50
I really didnt understand how this line equation helps my problem. Can you please elaborate this more?
– Frey
Aug 26 at 10:20
If your outer integral is over $theta$ and your inner integral is over $r,$ you need to get the bounds of $r$ along each radial line as a function of $theta.$ The upper bound is easy, because the integration along every radial line ends on the circle. But the integration starts on one of the lines $CQ$ or $BP,$ so you need to know how to compute the radius at which each radial line crosses one of those lines, as a function of the angle of the radial line. That function is just the function of the line in polar coordinates.
– David K
Aug 26 at 14:52
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I need to find the integral region $mathcal R$ which covers area CPRQ portion of the circle radius $t$.
The integration looks as
$$I=intint_mathcal Rf(r,theta)dr dtheta$$
Can someone help me to write these polar coordinate integration limits for $r$ and $theta$?
If necessary, he angle $RCQ$ is $beta$ where $Tan, beta=fracdh$.
The angle $ROQ$ is $alpha$ where
$cos,alpha = frach left(sqrtd^2+h^2 sqrtt^2-fracd^2 h^2d^2+h^2+d^2right)t left(d^2+h^2right)$
linear-algebra integration circle polar-coordinates
asked Aug 20 at 5:25
Frey
616312
I need to find the integral region $mathcal R$ which covers area CPRQ portion of the circle radius $t$.
The integration looks as
$$I=intint_mathcal Rf(r,theta)dr dtheta$$
Can someone help me to write these polar coordinate integration limits for $r$ and $theta$?
If necessary, he angle $RCQ$ is $beta$ where $Tan, beta=fracdh$.
The angle $ROQ$ is $alpha$ where
$cos,alpha = frach left(sqrtd^2+h^2 sqrtt^2-fracd^2 h^2d^2+h^2+d^2right)t left(d^2+h^2right)$
linear-algebra integration circle polar-coordinates
asked Aug 20 at 5:25
Frey
616312
asked Aug 20 at 5:25
Frey
616312
asked Aug 20 at 5:25
Frey
616312
asked Aug 20 at 5:25
Frey
616312
616312
Are you familiar with the equation for a line not through the origin in polar coordinates?
– David K
Aug 20 at 5:49
@David I am not much familiar but I have looked at some examples such as math.stackexchange.com/questions/2430448/… I am not sure how it can be applied in this case.
– Frey
Aug 21 at 7:13
Sadly, that question-asker was confused; the question did not actually involve polar coordinates. Try math.stackexchange.com/questions/1382437/… instead.
– David K
Aug 21 at 11:50
I really didnt understand how this line equation helps my problem. Can you please elaborate this more?
– Frey
Aug 26 at 10:20
If your outer integral is over $theta$ and your inner integral is over $r,$ you need to get the bounds of $r$ along each radial line as a function of $theta.$ The upper bound is easy, because the integration along every radial line ends on the circle. But the integration starts on one of the lines $CQ$ or $BP,$ so you need to know how to compute the radius at which each radial line crosses one of those lines, as a function of the angle of the radial line. That function is just the function of the line in polar coordinates.
– David K
Aug 26 at 14:52
add a comment |Â
Are you familiar with the equation for a line not through the origin in polar coordinates?
– David K
Aug 20 at 5:49
@David I am not much familiar but I have looked at some examples such as math.stackexchange.com/questions/2430448/… I am not sure how it can be applied in this case.
– Frey
Aug 21 at 7:13
Sadly, that question-asker was confused; the question did not actually involve polar coordinates. Try math.stackexchange.com/questions/1382437/… instead.
– David K
Aug 21 at 11:50
I really didnt understand how this line equation helps my problem. Can you please elaborate this more?
– Frey
Aug 26 at 10:20
If your outer integral is over $theta$ and your inner integral is over $r,$ you need to get the bounds of $r$ along each radial line as a function of $theta.$ The upper bound is easy, because the integration along every radial line ends on the circle. But the integration starts on one of the lines $CQ$ or $BP,$ so you need to know how to compute the radius at which each radial line crosses one of those lines, as a function of the angle of the radial line. That function is just the function of the line in polar coordinates.
– David K
Aug 26 at 14:52
Are you familiar with the equation for a line not through the origin in polar coordinates?
– David K
Aug 20 at 5:49
Are you familiar with the equation for a line not through the origin in polar coordinates?
– David K
Aug 20 at 5:49
@David I am not much familiar but I have looked at some examples such as math.stackexchange.com/questions/2430448/… I am not sure how it can be applied in this case.
– Frey
Aug 21 at 7:13
@David I am not much familiar but I have looked at some examples such as math.stackexchange.com/questions/2430448/… I am not sure how it can be applied in this case.
– Frey
Aug 21 at 7:13
Sadly, that question-asker was confused; the question did not actually involve polar coordinates. Try math.stackexchange.com/questions/1382437/… instead.
– David K
Aug 21 at 11:50
Sadly, that question-asker was confused; the question did not actually involve polar coordinates. Try math.stackexchange.com/questions/1382437/… instead.
– David K
Aug 21 at 11:50
I really didnt understand how this line equation helps my problem. Can you please elaborate this more?
– Frey
Aug 26 at 10:20
I really didnt understand how this line equation helps my problem. Can you please elaborate this more?
– Frey
Aug 26 at 10:20
If your outer integral is over $theta$ and your inner integral is over $r,$ you need to get the bounds of $r$ along each radial line as a function of $theta.$ The upper bound is easy, because the integration along every radial line ends on the circle. But the integration starts on one of the lines $CQ$ or $BP,$ so you need to know how to compute the radius at which each radial line crosses one of those lines, as a function of the angle of the radial line. That function is just the function of the line in polar coordinates.
– David K
Aug 26 at 14:52
If your outer integral is over $theta$ and your inner integral is over $r,$ you need to get the bounds of $r$ along each radial line as a function of $theta.$ The upper bound is easy, because the integration along every radial line ends on the circle. But the integration starts on one of the lines $CQ$ or $BP,$ so you need to know how to compute the radius at which each radial line crosses one of those lines, as a function of the angle of the radial line. That function is just the function of the line in polar coordinates.
– David K
Aug 26 at 14:52
add a comment |Â
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Are you familiar with the equation for a line not through the origin in polar coordinates?
– David K
Aug 20 at 5:49
@David I am not much familiar but I have looked at some examples such as math.stackexchange.com/questions/2430448/… I am not sure how it can be applied in this case.
– Frey
Aug 21 at 7:13
Sadly, that question-asker was confused; the question did not actually involve polar coordinates. Try math.stackexchange.com/questions/1382437/… instead.
– David K
Aug 21 at 11:50
I really didnt understand how this line equation helps my problem. Can you please elaborate this more?
– Frey
Aug 26 at 10:20
If your outer integral is over $theta$ and your inner integral is over $r,$ you need to get the bounds of $r$ along each radial line as a function of $theta.$ The upper bound is easy, because the integration along every radial line ends on the circle. But the integration starts on one of the lines $CQ$ or $BP,$ so you need to know how to compute the radius at which each radial line crosses one of those lines, as a function of the angle of the radial line. That function is just the function of the line in polar coordinates.
– David K
Aug 26 at 14:52