Trapezoidal Rule (1 Viewer)

[cutemouse]

Use the Trapezoidal rule with 5 function values to approximate and show that this approximation underestimates the value of the integral, without evaluating the integral by direct integration. [3 marks]

Thanks

 

[lychnobity]

I cbb finding the values, but use knowledge of lower bound limits.

You would be taking the lower bound limit and therefore, this would be underestimating the actual area.

 

[cyclicpermutate]

Use the Trapezoidal rule with 5 function values to approximate and show that this approximation underestimates the value of the integral, without evaluating the integral by direct integration. [3 marks]

Thanks

There are 2 ways to show that the approximation underestimates the integral. Firstly you could differentiate the function and show that it is concave down for the domain from 0 to 2. Or you could graph the logarithmic function and mearly show that its concave down for the aforementioned domain.

I believe that is the part of the problem that you dont understand, but just in case the trapizoidal rule is

 

[cutemouse]

Well the graph is given in the question... So I'd write something like "As the curve is concave down for 0<=x<=2, therefore the approx. under estimate the integral" ?

 

[cutemouse]

lower bound limit

Uhh, what exactly is that?

 

[lychnobity]

Uhh, what exactly is that?

The area that is lower than the actual area.

Look at the diagram, the rectangle formed by the y value of 'n' is the lower bound limit, because it's lower than the exact area.

Likewise with the rectangle formed by the y value of 'n+1', ie the upper bound limit

 

[cutemouse]

Uhh.. sorry if I'm not getting this.. But what exactly would I write in an exam if I got that question? (ie. how would I 'formally' show it?)

Thanks

 

[we0426]

Uhh.. sorry if I'm not getting this.. But what exactly would I write in an exam if I got that question? (ie. how would I 'formally' show it?)

Thanks

when u answer this question...u use ln..to find the values 4 the table..then its just using da formula...

this is not hard...try like questions that u have to drive the formula..shit those are crazy questions.....

 

[Uncle]

Another way to do this is to show your understanding that the trapezoidal rule approximates areas by using piecewise straight lines in the form of a₀ + a₁x which is identical to an area of a trapezium.

For each sub-interval, draw new piece-wise straight lines, there should be a picture of this, if you look carefully or better yet zoom into it, the blue lines should be under the red curve of the actual function complimenting the explanation.

Trapezoidal rule becomes important in second-year engineering mathematics and chances are most possible functions cannot be evaulated by direct integration (a.k.a. analytically).