Approximation Help Please: Consider ∫ [4 to 9] (1/√x) dx 1); Find n for which...

[AmbR]

Approximation Help Please: Consider ∫ [4 to 9] (1/√x) dx 1); Find n for which...

[FONT=&quot]Consider ∫ [4 to 9] (1/√x) dx [/FONT]

[FONT=&quot]1) Find a value of n for which |Ln − Rn| ≤ 0.1 [/FONT]

[FONT=&quot]2) Use that value of n to find Ln and Rn. [/FONT]

[FONT=&quot]3) If you take the average of the left- and right-hand sums, will your approximation be larger than the integral, or smaller? [/FONT]

[FONT=&quot]4) Compare your numerical approximations to the answers you get using the Fundamental Theorem of Calculus.


So far this is what I have so far:
The difference of areas is Ln-Rn = [(b-a)/n](F(a)-f(b))Ln-Rn = (b-a)/n•(f(a)-f(b))
Then I got: f(a) - f(b) = 1/6•f(a)-f(b) = 1/6 & b-a =5
5/6(1/n) [/FONT][FONT=&quot]≤ 1/10
[/FONT][FONT=&quot]so n = 9
[/FONT][FONT=&quot]
But I don't know how to solve #2 & #3.[/FONT]

 

[Deleted member 4993]

Consider ∫ [4 to 9] (1/√x) dx

1) Find a value of n for which |Ln − Rn| ≤ 0.1

2) Use that value of n to find Ln and Rn.

3) If you take the average of the left- and right-hand sums, will your approximation be larger than the integral, or smaller?

4) Compare your numerical approximations to the answers you get using the Fundamental Theorem of Calculus.

What method/s of Numerical Integration have you been taught?

 

[stapel]

Consider ∫ [4 to 9] (1/√x) dx

1) Find a value of n for which |Ln − Rn| ≤ 0.1

What is the definition (at this stage, in your particular textbook) of (what I'm assuming are) L_n and R_n? Are they perhaps the expressions for the right-hand-endpoint and left-hand-endpoint Reimann sums, using equal-width subdivisions? If so, please confirm; if not, please correct with specifics.

So far this is what I have so far:
The difference of areas is Ln-Rn = [(b-a)/n](F(a)-f(b))Ln-Rn = (b-a)/n•(f(a)-f(b))

Taking that first equality, you have the following:

. . . . .\(\displaystyle L_n\, -\, R_n = \left(\dfrac{b\, -\, a}{n}\right)\, \big[F(a)\, -\, f(b)\big]\, L_n\, -\, R_n\)

This reduces to:

. . . . .\(\displaystyle 1\, =\, \left(\dfrac{b\, -\, a}{n}\right)\, \big[F(a)\, -\, f(b)\big]\)

...assuming that L_n does not equal zero. Was this what you meant? If so, how did you get this? If not, what did you mean?

Also, how are you defining the two functions, "F(x)" and "f(x)"?

I'm not sure how you got the rest of what you posted...?

2) Use that value of n to find Ln and Rn.

3) If you take the average of the left- and right-hand sums, will your approximation be larger than the integral, or smaller?

4) Compare your numerical approximations to the answers you get using the Fundamental Theorem of Calculus.

But I don't know how to solve #2 & #3.

#2) You have to have found the left- and right-hand sums (or, at least, the expressions for them) in part #1. Plug the value you got for n into those formulas.

#3) Add the two sums. Divide by 2. Compare.