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1600dave

Member

Registered: 2006-10-10

Posts: 4

solving definite integrals, solving systems uisng Gaussian elimination

not sure if this is the correct section to post but seemed most suitable to me...please move if needed.

here's the problem integral problem

the way i've been doing it, and the way i want to is to find the antiterivative and then sub in the limits, however, have some trouble doing so.
i'l use '~' as the integral symbol
and 'sqrt' as square root symbol

first i let u = sinx   where du=cosxdx
therefore

=-sinx 0~1/2 sqrt(u)*du       if x=0, u=0      if x=1/2, u=0.009

= -sinx 0~0.009 u^1/2*du

=-sinx [2/3U^2/3]

=[-sinx(2/3sinx^3/2)]

and the answers are completely wrong. i have check it on my calculator and it should be 0.031 or thereabouts.

secondly, solving the systems using the Gaussian elimination method, i haven't been able to come across any results that satisfy and of the equations.

here is the equations giving me trouble

and here is what i worked out

  2 4 -6, 2           1 0 4, 6         1 0 4, 6           1 0 4, 6          1 0 0,-98
[ 0 1 2,  4 ]  ~  [ 0 1 2, 4 ]  ~  [0 1 2, 4 ]  ~  [ 0 1 2, 4 ]  ~  [0 1 0, 75/22]
  1 0 4, -6           2 4 -6, 2        0 4 -14, -10     0 0 -22, -26    0 0 1, 13/11

if someone could smash up what they thinks right, thatd be awesome as i'm just blind to what i've done wrong (most people are to their own mistakes!), or how i should go about it. these are the methods i need to use to solve them to so please dont smash up easier ways, i'd be using them if i could!
thanks,
dave

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: solving definite integrals, solving systems uisng Gaussian elimination

For the second one, I used row reduction to get this far:

2  0  -14   |  -14
0  1    2    |  4
0  0   22   |  2

And from this I get:

Which checks when substituted back into the system.

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: solving definite integrals, solving systems uisng Gaussian elimination

For the integral:

Notice that

And here it just so happens that cos(x) is the derivative of sin(x).

So we get:

And now the definite integral is easy to evaluate by the fundamental theorem of calculus.

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: solving definite integrals, solving systems uisng Gaussian elimination

Oh I see you have almost that, but with an extra -sin(x), which shouldn't be there.

1600dave

Member

Registered: 2006-10-10

Posts: 4

Re: solving definite integrals, solving systems uisng Gaussian elimination

so, is the answer i had roughly calculated before of approx 0.031 correct, as when i'm solving it now i'm getting answers that are like 5.4x10^-3

1600dave

Member

Registered: 2006-10-10

Posts: 4

Re: solving definite integrals, solving systems uisng Gaussian elimination

alll sorted, had my calc in degree mode not radians. answer is 0.221

thanks everyone!!