solving definite integrals, and solving systems using the Gaussian Elimination method

[1600dave]

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




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

[Glaysher]

Integral substitution u = sin x

du/dx = cos x

dx = 1/[cos x] du

x = 0, u = 0
x = 1/2, u = sin (1/2)

Integral becomes

Integral between 0 and sin (1/2) of u^(1/2) cos x (1/cos x) du

Integral between 0 and sin (1/2) of u^(1/2) du

[(2/3)u^(3/2)] evaluated between 0 and sin (1/2)

Not sure where your - sin x came from?

[Glaysher]

2 4 -6 2
0 1 2 4
1 0 4 -6

1 2 -3 1
0 1 2 4
0 -2 7 -7

1 0 -7 -7
0 1 2 4
0 0 11 1

1 0 0 (-6 4/11)
0 1 0 (3 9/11)
0 0 1 (1/11)

x₁ = -6 4/11
x₂ = 3 9/11
x₃ = 1/11

[1600dave]

Integral substitution u = sin x

du/dx = cos x

dx = 1/[cos x] du

x = 0, u = 0
x = 1/2, u = sin (1/2)

Integral becomes

Integral between 0 and sin (1/2) of u^(1/2) cos x (1/cos x) du

Integral between 0 and sin (1/2) of u^(1/2) du

[(2/3)u^(3/2)] evaluated between 0 and sin (1/2)

Not sure where your - sin x came from?

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]

ahh the answer i had calculated before using my calculator shouldnt be 0.031, as the calculator was set in degree mode. now its in radians i'm getting 0.221., as expected.
thanks guys all works now