length of equation using matlab

Can anyone please help me with this

y=x^3/25-3*x^2/625-2*x/15625;

I want to find the length of this equation using matlab. Thanks

Answer 1

[Walter Roberson]

There are multiple ways of expressing the same equation, so what does it mean to take the "length" of the equation?

Take your pick:

y = x*(625*x^2-75*x-2)/15625
y = x*(x*(x/25-3/625)-2/15625)
y = x^3/25-3*x^2/625-2*x/15625
`=`(y, `+`(`*`(Fraction(1,25),`^`(x,3)), `*`(Fraction(-3,625), `^`(x,2)), `*`(Fraction(-2,15625),x)))

Perhaps what you are looking for is:

length('y=x^3/25-3*x^2/625-2*x/15625')

[rsch tosh]

Thanks for the reply. But What I am looking for is length of the arc, that is created by this equation. mathematically it is given by integral of squareroot of (1+(dy/dx)^2). The integral is between 0 to x.

I want a matlab code that can do it.

Thanks a lot.

Answer 2

[Walter Roberson]

Well, if you are sure, then let x1 be the upper limit of integration, then define

EllipticE = @(z,k) feval(symengine, ellipticE, z, k);
EllipticF = @(z, k) feval(symengine, ellipticF, z, k);

and the integral is

(1/1609325600097658950 + 125/64373024003906358 * i) * ((-6 * (46875 + 15 *
i + i * (15 - 46875 * i) ^ (1/2) * (15 + 46875 * i) ^ (1/2)) * (75 * x1 - 3 +
(15 + 46875 * i) ^ (1/2)) / (-75 * x1 + 3 + (15 - 46875 * i) ^ (1/2))) ^
(1/2) * (-((15 + 46875 * i) ^ (1/2) - (15 - 46875 * i) ^ (1/2)) * (75 *
x1 - 3 + (15 - 46875 * i) ^ (1/2)) / (-75 * x1 + 3 + (15 - 46875 * i) ^
(1/2))) ^ (1/2) * ((((-2/5859375 + 2/1875 * i) + (2/234375 - 2/75 * i) *
x1) * (15 + 46875 * i) ^ (1/2) + 48828133/5859375 + 1/625 * i + (1/3125
+ i) * x1 ^2 + (-2/78125 - 2/25 * i) * x1) * (15 - 46875 * i) ^ (1/2) +
(16276039/1953125 + 13/1875 * i + (-1/3125 + i) * x1 ^2 + (2/78125 - 2/25 *
i) * x1) * (15 + 46875 * i) ^ (1/2) + 19531252/390625 - 19531252/15625 *
x1) * 244140629 ^ (1/2) * ((15 - 46875 * i) ^ (1/2) * (-75 * x1 + 3 +
(15 + 46875 * i) ^ (1/2)) / (-75 * x1 + 3 + (15 - 46875 * i) ^ (1/2))) ^
(1/2) * EllipticF(125 * 2 ^ (1/2) * 3 ^ (1/2) * (i * (75 * x1 - 3 + (15 -
46875 * i) ^ (1/2)) / (75 * x1 - 3 - (15 - 46875 * i) ^ (1/2))) ^ (1/2) /
(30 + 30 * 2 ^ (1/2) * 4882813 ^ (1/2)) ^ (1/2), 1/3125 * i + (1/46875 *
i) * (15 - 46875 * i) ^ (1/2) * (15 + 46875 * i) ^ (1/2)) - 2/5 * (-6 *
(46875 + 15 * i + i * (15 - 46875 * i) ^ (1/2) * (15 + 46875 * i) ^ (1/2)) *
(75 * x1 - 3 + (15 + 46875 * i) ^ (1/2)) / (-75 * x1 + 3 + (15 - 46875 *
i) ^ (1/2))) ^ (1/2) * (-((15 + 46875 * i) ^ (1/2) - (15 - 46875 * i) ^
(1/2)) * (75 * x1 - 3 + (15 - 46875 * i) ^ (1/2)) / (-75 * x1 + 3 + (15 -
46875 * i) ^ (1/2))) ^ (1/2) * (((-1/1171875 + 1/46875 * x1) * (15 + 46875
* i) ^ (1/2) - 1/15625 * x1 + 4/1171875 - 1/150 * i + 1/1250 * x1 ^2) *
(15 - 46875 * i) ^ (1/2) + (-1/1250 * x1 ^2 + 1/15625 * x1 - 4/1171875 +
1/150 * i)) * (15 + 46875 * i) ^ (1/2) + 1/78125 - 1/25 * i + (-1/ 3125 +
i) * x1) * 244140629 ^ (1/2) * ((15 - 46875 * i) ^ (1/2) * (-75 * x1 + 3 +
(15 + 46875 * i) ^ (1/2)) / (-75 * x1 + 3 + (15 - 46875 * i) ^ (1/2)))
^ (1/2) * EllipticE(125 * 2 ^ (1/2) * 3 ^ (1/2) * (i * (75 * x1 - 3 +
(15 - 46875 * i) ^ (1/2)) / (75 * x1 - 3 - (15 - 46875 * i) ^ (1/2))) ^
(1/2) / (30 + 30 * 2 ^ (1/ 2) * 4882813 ^ (1/2)) ^ (1/2), 1/ 3125 * i +
((1/46875) * i) * (15 - 46875 * i) ^ (1/2) * (15 + 46875 * i) ^ (1/2)) - 2 *
(((-8/15625 + i) + 2/15625 * (15 + 46875 * i) ^ (1/2)) * (15 - 46875 * i) ^
(1/2) + (-6/3125 + 6 * i) + (8/15625 - i) * (15 + 46875 * i) ^ (1/2)) *
((15 - 46875 * i) ^ (1/2) * (-3 + (15 + 46875 * i) ^ (1/2)) / (((15 +
46875 * i) ^ (1/2) - (15 - 46875 * i) ^ (1/2)) * (3 + (15 - 46875 * i) ^
(1/2)))) ^ (1/2) * ((15 - 46875 * i) ^ (1/2) * (3 + (15 + 46875 * i) ^
(1/2)) / (3 + (15 - 46875 * i) ^ (1/2))) ^ (1/2) * (3515625 * x1 ^4 -
562500 * x1 ^3 + 15000 * x1 ^2 + 600 * x1 + 244140629) ^ (1/2) * ((-(15 +
46875 * i) ^ (1/2) + (15 - 46875 * i) ^ (1/2)) * (-3 + (15 - 46875 * i) ^
(1/2)) / (3 + (15 - 46875 * i) ^ (1/2))) ^ (1/2) * EllipticE(((3 * (15 +
46875 * i) ^ (1/2) - (15 + 46875 * i) ^ (1/2) * (15 - 46875 * i) ^ (1/2) -
3 * (15 - 46875 * i) ^ (1/2) + 15 - 46875 * i) / (3 * (15 + 46875 * i) ^
(1/2) + (15 + 46875 * i) ^ (1/2) * (15 - 46875 * i) ^ (1/2) + 3 * (15 -
46875 * i) ^ (1/2) + 15 - 46875 * i)) ^ (1/2), ((15 + 46875 * i) ^ (1/2) +
(15 - 46875 * i) ^ (1/2)) / ((15 + 46875 * i) ^ (1/2) - (15 - 46875 * i)
^ (1/2))) - 4/5 * ((15 - 46875 * i) ^ (1/2) * (-3 + (15 + 46875 * i) ^
(1/2)) / (((15 + 46875 * i) ^ (1/2) - (15 - 46875 * i) ^ (1/2)) * (3 +
(15 - 46875 * i) ^ (1/2)))) ^ (1/2) * ((15 - 46875 * i) ^ (1/2) * (3 +
(15 + 46875 * i) ^ (1/2)) / (3 + (15 - 46875 * i) ^ (1/2))) ^ (1/2) *
((48828133 / 6250 + (3 / 2) * i + (-1 / 3125 + i) * (15 + 46875 * i) ^
(1/2)) * (15 - 46875 * i) ^ (1/2) + 29296878 / 625 + (48828117 / 6250 +
(13 / 2) * i) * (15 + 46875 * i) ^ (1/2)) * (3515625 * x1^4 - 562500 *
x1^3 + 15000 * x1^2 + 600 * x1 + 244140629) ^ (1/2) * ((-(15 + 46875 * i)
^ (1/2) + (15 - 46875 * i) ^ (1/2)) * (-3 + (15 - 46875 * i) ^ (1/2)) /
(3 + (15 - 46875 * i) ^ (1/2))) ^ (1/2) * EllipticF(((3 * (15 + 46875 *
i) ^ (1/2) - (15 + 46875 * i) ^ (1/2) * (15 - 46875 * i) ^ (1/2) - 3 *
(15 - 46875 * i) ^ (1/2) + 15 - 46875 * i) / (3 * (15 + 46875 * i) ^
(1/2) + (15 + 46875 * i) ^ (1/2) * (15 - 46875 * i) ^ (1/2) + 3 * (15 -
46875 * i) ^ (1/2) + 15 - 46875 * i)) ^ (1/2), ((15 + 46875 * i) ^ (1/2)
+ (15 - 46875 * i) ^ (1/2)) / ((15 + 46875 * i) ^ (1/2) - (15 - 46875 *
i) ^ (1/2))) + ((3 / 3125) * (15 - 46875 * i) ^ (1/2) * (x1 - 1/25) *
(858306898828125 * x1^4 - 137329103812500 * x1^3 + 3662109435000 * x1 ^2 +
146484377400 * x1 + 59604646728515641) ^ (1/2) + (-292968762/3125 - 18 * i) +
(732421947/78125 + 6 * i) * (15 - 46875 * i) ^ (1/2)) * (3515625 * x1^4 -
562500 * x1^3 + 15000 * x1^2 + 600 * x1 + 244140629) ^ (1/2) + 11250 *
(1/75 * (-3/25 * x1^2 + 6/625 * x1 + 2/15625 + i) * (x1 - 1/25) * (15 -
46875 * i) ^ (1/2) + 48828127/5859375 + 1/625 * i + (-1/3125 + i) * x1^2 +
(2/78125 - (2/25) * i) * x1) * 244140629 ^ (1/2)) * (15 - 46875 * i) ^
(1/2) * 244140629 ^ (1/2) / (3515625 * x1 ^4 - 562500 * x1 ^3 + 15000 *
x1 ^2 + 600 * x1 + 244140629) ^ (1/2)

Or in more readable form but likely slightly less accurate form,

t1 = sqrt((15+46875*i));
t2 = sqrt((15-46875*i));
t3 = t2 * t1;
t5 = 75 * x1;
t8 = -t5 + 3 + t2;
t9 = 0.1e1 / t8;
t12 = sqrt(-6 * t9 * (t5 - 3 + t1) * ((46875+15*i) + (i) * t3));
t13 = t1 - t2;
t14 = t5 - 3 + t2;
t17 = sqrt(-t9 * t14 * t13);
t18 = t17 * t12;
t22 = x1 ^ 2;
t27 = (-0.1e1 / 0.3125e4+1*i) * t22;
t28 = (0.2e1 / 0.78125e5-0.2e1 / 0.25e2*i) * x1;
t34 = sqrt(0.244140629e9);
t38 = sqrt(t9 * (-t5 + 3 + t1) * t2);
t39 = t38 * t34;
t40 = sqrt(0.2e1);
t41 = sqrt(0.4882813e7);
t47 = sqrt(0.3e1);
t50 = sqrt((-1*i) / t8 * t14);
t53 = (0.25e2 / 0.6e1) * t50 * t47 * t40 * sqrt(0.30e2) * ((0.1e1 + t41 * t40) ^ (-0.1e1 / 0.2e1));
t56 = (0.1e1 / 0.3125e4*i) + (0.1e1 / 0.46875e5*i) * t1 * t2;
t57 = EllipticF(t53, t56);
t63 = x1 / 0.15625e5;
t64 = t22 / 0.1250e4;
t72 = EllipticE(t53, t56);
t83 = 0.1e1 / t13;
t85 = 0.1e1 / (3 + t2);
t88 = sqrt(t85 * t83 * t2 * (-3 + t1));
t93 = sqrt(t85 * (3 + t1) * t2);
t95 = t22 ^ 2;
t97 = t22 * x1;
t102 = sqrt((3515625 * t95 - 562500 * t97 + 15000 * t22 + 600 * x1 + 244140629));
t106 = sqrt(-t85 * (-3 + t2) * t13);
t107 = t106 * t102;
t108 = 3 * t1;
t109 = 3 * t2;
t114 = sqrt(0.1e1 / (t108 + t3 + t109 + (15-46875*i)) * (t108 - t3 - t109 + (15-46875*i)));
t116 = t83 * (t1 + t2);
t117 = EllipticE(t114, t116);
t128 = EllipticF(t114, t116);
t132 = x1 - 0.1e1 / 0.25e2;
t139 = sqrt((858306898828125 * t95 - 137329103812500 * t97 + 3662109435000 * t22 + 146484377400 * x1 + 59604646728515641));
t159 = (0.1e1 / 0.1609325600097658950e19+0.125e3 / 0.64373024003906358e17*i) * t34 * t2 / t102 * (t57 * t39 * (t2 * (t1 * ((-0.2e1 / 0.5859375e7+0.2e1 / 0.1875e4*i) + (0.2e1 / 0.234375e6-0.2e1 / 0.75e2*i) * x1) + (0.48828133e8 / 0.5859375e7+0.1e1 / 0.625e3*i) + (0.1e1 / 0.3125e4+1*i) * t22 + (-0.2e1 / 0.78125e5-0.2e1 / 0.25e2*i) * x1) + t1 * ((0.16276039e8 / 0.1953125e7+0.13e2 / 0.1875e4*i) + t27 + t28) + (0.19531252e8 / 0.390625e6) - (0.19531252e8 / 0.15625e5 * x1)) * t18 - (0.2e1 / 0.5e1) * t72 * t39 * (t2 * (t1 * (-0.1e1 / 0.1171875e7 + x1 / 0.46875e5) - t63 + (0.4e1 / 0.1171875e7-0.1e1 / 0.150e3*i) + t64) + t1 * (-t64 + t63 + (-0.4e1 / 0.1171875e7+0.1e1 / 0.150e3*i)) + (0.1e1 / 0.78125e5-0.1e1 / 0.25e2*i) + (-0.1e1 / 0.3125e4+1*i) * x1) * t18 - 2 * t117 * t107 * t93 * t88 * (t2 * ((-0.8e1 / 0.15625e5+1*i) + (0.2e1 / 0.15625e5) * t1) + (-0.6e1 / 0.3125e4+6*i) + (0.8e1 / 0.15625e5-i) * t1) - (0.4e1 / 0.5e1) * t128 * t107 * (t2 * ((0.48828133e8 / 0.6250e4+0.3e1 / 0.2e1*i) + (-0.1e1 / 0.3125e4+1*i) * t1) + (0.29296878e8 / 0.625e3) + (0.48828117e8 / 0.6250e4+0.13e2 / 0.2e1*i) * t1) * t93 * t88 + t102 * ((0.3e1 / 0.3125e4) * t139 * t132 * t2 + (-0.292968762e9 / 0.3125e4-18*i) + (0.732421947e9 / 0.78125e5+6*i) * t2) + 11250 * t34 * (t2 * t132 * (-(0.3e1 / 0.25e2 * t22) + (0.6e1 / 0.625e3 * x1) + (0.2e1 / 0.15625e5+1*i)) / 75 + (0.48828127e8 / 0.5859375e7+0.1e1 / 0.625e3*i) + t27 + t28));