How do I solve this integral using a) the midpoint rule and b) simp son's rule to approximate the integral? can you please explain? thank you!!

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How do I solve this integral using a) the midpoint rule and b) simp son's rule to approximate the integral? can you please explain? thank you!!

Mathematics
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\[\int\limits_{0}^{2} \frac{ x }{ 1+x^2 } dx \]
where n = 10
its finding an approximation to the area under the curve ...

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yes:) how can i set it up? I'm a bit confused with the process :o
instead of left or right values, you are grabbing the value at the middle of the interval
ooh okay! yes:) what do we do to start off?
im not sure how easy it would be to construct the rule .. your material should have it
okay! then, could you please show me how id go about solving this integral?
the hardest part is constructing the rule .. since i dont recall the formula for it off hand. do you have the formula for it?
|dw:1443474213664:dw|
yes! it is this for midpoint rule integral of a to b f(x) dx is approximately M_n = delta x [f(x1)+f(x2)+….+f(xn)] where delta x = b-a / n and xi=1/2(xi-1 + xi) = midpoint of [xi-1, xi]
i never heard of this but i watched this video and I understand it but it is a lot of work https://www.youtube.com/watch?v=fKtSR0_xf5g
powers fluctuating over here ... so my connection isnt stable
okay, i see!!
f (a + ((i_n)dx+(i_(n+1)dx)/2 ) f (a + ((i_n)+(i_(n+1)) dx/2 ) , for i from a to n-1
okay!
\[\sum_{i=0}^{n-1}f(\color{red}{a+\frac12\left( i\frac{b-a}{n}+(i+1)\frac{b-a}{n}\right)})\] \[\sum_{i=0}^{n-1}f(\color{red}{a+\frac{b-a}{2n}\left( 2i+1\right)})\] does that seem plausible?
a=0, b=2 sooo \[\sum_{i=0}^{9}f(\color{red}{\frac{2i+1}{10}})\frac1{10}\]
x= 1/10 3/10 5/10 7/10 9/10 11/10 13/10 15/10 17/10 19/10
determine f(x) for each value of x ....
simpsons rule is using a parabolic shape to define the points of reference with.

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