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.
DEFDBL AZ
pi = ATN(1) * 4
h = .0000001
y0 = LOG(4 / 3)
yb = y0
CLS
x = yb
t = 0
DO
t = t + h / SQR(EXP(x)  1)
IF t > pi / 6 THEN EXIT DO
tprev = t
x = x + h
LOOP
PRINT x  h, tprev, tprev  pi / 6
PRINT x, t, t  pi / 6
Finds that
y integral diff from pi/6
.6931469771900832 .5235987088310166 6.676728229554297D08
.6931470771900844 .5235988088310281 3.32327291981409D08
so a good approximation for y would be .693147077190084, which is close enough to ln(2)~=.6931471805599453 to take the latter as probably being the answer. The discrepancy from .6931471805599453 is probably attributable to the coarseness of h in the Riemann sum.

Posted by Charlie
on 20071127 10:43:27 