Approximation and Simpson's Rule

[ada0713]

Homework Statement

Suppose the exact value of a particular definite integral is 6. The following questions refer to estimates of this integral using the left, trapezoid, and Simpson's rules. Use what you know about approximate errors to answer the following questions. Give your answer to 4 decimal places.

(a) Suppose Left(40)=2.3055. Estimate Left(120)


(b) Suppose Trap(40)=2.7250. Estimate Trap(120)


(c) Suppose Simp(40)=3.1680. Estimate Simp(120)

The Attempt at a Solution

I know that Actual- Approximated= Error, but I am not sure how to use this concept
to solve the problem. I've tried to find the ratio of each, but not enough information
was given in order to estimate what it asked for. (In terms of "not enough information",
I mean--- Shouldn't they give us the actual function whose definite integral is 6?)
Please help me with the start and I'll try to figure out he rest!

 

[ada0713]

Anyone know how to do this problem? HELP!

 

[Vid]

Error in trap(40) = 6 - 2.7250 = K2(b-a)^3/(12)(40)^2. Therefore, K2(b-a)^3/12 = 40^2(6-2.7250).
Then trap(120) = 40^2(6-2.7250)*(1/(120)^2

 

[ada0713]

What does K2 stand for? I don't quite understand how you did it..

 

[Vid]

\left|E_{T}\right| = \frac{K_{2}(b-a)^{3}}{12n^{2}}

Where K_{2} \geq \left|f''(x)\right| on [a,b]

Since we know the exact value of the integral, we can subtract the approximation given to get a value for the error of T(40). In this case 6 - 2.7250 = 3.275.

3.275 =\frac{K_{2}(b-a)^{3}}{12(40)^{2}}

Multiply both sides by 40^2.

(40^2)(3.275) = \frac{K_{2}(b-a)^{3}}{12}

Now, use this fact to find the error when n = 120.

 

[ada0713]

Oh.. Thanks..
But it seems the formula up there is only useful when we find the approximation for
trapezoide. Are there specific formulas for left rule and simpsons's rule as well?
This is new to me so I tried to find the concept in the textbook but I don't think it mentions this :(

 

[NateTG]

I'm not sure what your text is like. I saw those error formulae in second-year calculus.

http://en.wikipedia.org/wiki/Rectangle_rule
http://en.wikipedia.org/wiki/Simpson's_rule