the question is as follow:
The partition does not always have to be equal intervals. Consider evaluating f(x)=x3 between 3 and 5, but splitting up the interval into a partition in which the end points of the subintervals are in a geometric progression. The common ratio r has to be chosen so that 3 is the first term and 5 is the last. Also the subintervals must be capable of getting smaller as n the number of subintervals increases. Check that the geometric series
a, ar, ar2, ar3,.....ari, .....arn =b
with r= and suitable choices for a and b satisfies these criteria. Treating the difference between ari and ar(i-1) as the width of the subinterval and using the right hand endpoint of the subinterval, evaluate the Riemann sum to n terms for f(x)=x3. Find the limit as n tends to infinity to show that the partition does not affect the result.
here is what i have got so far, can anyone check if im doing it right? thanks
>for i from 0 to 5 do a*r^i end do; -> a list of number appear in sequence ie:3, 3.157...,3.323...3.497...etc
>evalf(sum(f(xj^*)dxj,i=1..100)) -> my value is sth like 162.4788870...
I tried to find the limit, but maple 16 freezed so i think i must have done sth seriously wrong?
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo>≔</mo><mi>a</mi><mo>⋅</mo><msup><mi>r</mi><mrow><mn>10</mn></mrow></msup><mo>;</mo><mo> </mo></mrow></math>