Question
Use the trapezoidal rule with n = 8 to approximate
∫4,1 cos(x)/x dx
∫4,1 cos(x)/x dx
Answers
Answered by
Tammy
Wow, this is a hard question
first of all, in the domain from 1 to 4, the graph crosses the x-axis at π/2
so we will not be able to cut it into equal trapezoids all on one side of the x-axis.
secondly, if we wanted to actually integrate cosx/x, we will have great difficulty, integration by parts doesn't work, trig substituion doesn't work, no regular methods will work.
we will have to expand cosx = 1 - x^2/2! + x^4/4! - x^6/6! + ... then divide each term by x. We then get a sequence which can be integrated and then
differentiated ending up with an infinite series.
The bot's answer is of course incorrect
first of all, in the domain from 1 to 4, the graph crosses the x-axis at π/2
so we will not be able to cut it into equal trapezoids all on one side of the x-axis.
secondly, if we wanted to actually integrate cosx/x, we will have great difficulty, integration by parts doesn't work, trig substituion doesn't work, no regular methods will work.
we will have to expand cosx = 1 - x^2/2! + x^4/4! - x^6/6! + ... then divide each term by x. We then get a sequence which can be integrated and then
differentiated ending up with an infinite series.
The bot's answer is of course incorrect
Answered by
oobleck
with 8 intervals, we have to evaluate f(x) at nine points: 1+k/3 for k=0..8
so the value will be
1/6 (f(1) + 2f(4/3) + ... + 2f(11/3) + f(4))
so now just plug and chug.
Or, use any of various handy trapezoidal rule calculators online.
so the value will be
1/6 (f(1) + 2f(4/3) + ... + 2f(11/3) + f(4))
so now just plug and chug.
Or, use any of various handy trapezoidal rule calculators online.
Answered by
Bot
The approximate value is 0.845.