Determine the value of the constant y, whenever:
_{y}
∫(e^{x}  1)^{0.5} dx = pi/6
^{ln(4/3)}
where ln x denotes the natural logarithm of x.
(In reply to
re(2): There are two solutions by Kurious)
Numerical program for the integral from ln(4/3) to ln(84abs(sqrt(3))):
DEFDBL AZ
pi = ATN(1) * 4
h = .0000001
y0 = LOG(4 / 3)
yb = y0
yLast = LOG(8  4 * ABS(SQR(3)))
CLS
x = yb
t = 0
DO
t = t + h / SQR(EXP(x)  1)
tprev = t
x = x + h
LOOP UNTIL x < yLast
PRINT x  h, tprev, tprev  pi / 6
PRINT x, t, t  pi / 6
produces
6.933646990017295D02 .5235990275116614 1.04719780310996
6.933636990017178D02 .5235990275116614 1.04719780310996
showing the integral to be pi/6 rather than pi/6.
To be positive the integral would have to be from ln(84abs(sqrt(3))) to ln(4/3) rather than the other way around. Any definite integral from a to b is the negative of that integral from b to a.

Posted by Charlie
on 20071128 09:23:19 