## anonymous one year ago 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!!

1. anonymous

$\int\limits_{0}^{2} \frac{ x }{ 1+x^2 } dx$

2. anonymous

where n = 10

3. amistre64

its finding an approximation to the area under the curve ...

4. anonymous

yes:) how can i set it up? I'm a bit confused with the process :o

5. amistre64

instead of left or right values, you are grabbing the value at the middle of the interval

6. anonymous

ooh okay! yes:) what do we do to start off?

7. amistre64

im not sure how easy it would be to construct the rule .. your material should have it

8. anonymous

okay! then, could you please show me how id go about solving this integral?

9. amistre64

the hardest part is constructing the rule .. since i dont recall the formula for it off hand. do you have the formula for it?

10. amistre64

|dw:1443474213664:dw|

11. anonymous

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]

12. anonymous

@amistre64 @IrishBoy123 ?

13. zpupster

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

14. amistre64

powers fluctuating over here ... so my connection isnt stable

15. anonymous

okay, i see!!

16. amistre64

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

17. anonymous

okay!

18. amistre64

$\sum_{i=0}^{n-1}f(\color{red}{a+\frac12\left( i\frac{b-a}{n}+(i+1)\frac{b-a}{n}\right)})$ $\sum_{i=0}^{n-1}f(\color{red}{a+\frac{b-a}{2n}\left( 2i+1\right)})$ does that seem plausible?

19. amistre64

a=0, b=2 sooo $\sum_{i=0}^{9}f(\color{red}{\frac{2i+1}{10}})\frac1{10}$

20. amistre64

x= 1/10 3/10 5/10 7/10 9/10 11/10 13/10 15/10 17/10 19/10

21. amistre64

determine f(x) for each value of x ....

22. amistre64

simpsons rule is using a parabolic shape to define the points of reference with.