Calculating Right Endpoint Riemann Sum

[Hckyplayer8]

Calculate R₄ right endpoint approximation, for f(x) = x² + x for the interval [0,2]

So R₄ refers to how many partitions I am looking to break this area into. Thus I'll be going with four rectangles over the aforementioned interval. In order to fit four rectangles over the interval [0,2] each delta x must be 1/2.

I found f(x) for 0, 1/2, 1, 3/2 and 2 which ended up being 0, 3/4, 2, 15/4 and 6.

Lastly I found the area of the rectangles by multiplying 1/2 against the summation of the widths. My final answer was 25/4 which will be an overestimate because right endpoints on an increasing function will account for addition area above the curve.

Do this seem reasonable?

 

[Steven G]

Calculate R₄ right endpoint approximation, for f(x) = x² + x for the interval [0,2]

So R₄ refers to how many partitions I am looking to break this area into. Thus I'll be going with four rectangles over the aforementioned interval. In order to fit four rectangles over the interval [0,2] each delta x must be 1/2.

I found f(x) for 0, 1/2, 1, 3/2 and 2 which ended up being 0, 3/4, 2, 15/4 and 6.

Lastly I found the area of the rectangles by multiplying 1/2 against the summation of the widths. My final answer was 25/4 which will be an overestimate because right endpoints on an increasing function will account for addition area above the curve.

Do this seem reasonable?

You have the correct answer but you do have a major typo (in my opinion). It is not a number but a word. Can you find it?

 

[Dr.Peterson]

Calculate R₄ right endpoint approximation, for f(x) = x² + x for the interval [0,2]

So R₄ refers to how many partitions I am looking to break this area into. Thus I'll be going with four rectangles over the aforementioned interval. In order to fit four rectangles over the interval [0,2] each delta x must be 1/2.

I found f(x) for 0, 1/2, 1, 3/2 and 2 which ended up being 0, 3/4, 2, 15/4 and 6.

Lastly I found the area of the rectangles by multiplying 1/2 against the summation of the widths. My final answer was 25/4 which will be an overestimate because right endpoints on an increasing function will account for addition area above the curve.

Do this seem reasonable?

Yes, except ...

I don't think you mean "the summation of the widths", but of the heights. Right? The width, as I think of it, is 1/2.

More important, you aren't really adding all five numbers you listed, but only the four right end values: 3/4, 2, 15/4, 6. If the first had not been 0, then this would make a difference!

 

[Hckyplayer8]

Yes, except ...

I don't think you mean "the summation of the widths", but of the heights. Right? The width, as I think of it, is 1/2.

More important, you aren't really adding all five numbers you listed, but only the four right end values: 3/4, 2, 15/4, 6. If the first had not been 0, then this would make a difference!

Yes. My mistake. Heights...the y values.

Also I see what you mean regarding incorrectly including point 0,0 in the assessment.

Thank you both for commenting!

 

[Steven G]

Yep, that was the typo I was thinking about!

 

[pka]

Calculate R₄ right endpoint approximation, for f(x) = x² + x for the interval [0,2]
So R₄ refers to how many partitions I am looking to break this area into. Thus I'll be going with four rectangles over the aforementioned interval. In order to fit four rectangles over the interval [0,2] each delta x must be 1/2.
I found f(x) for 0, 1/2, 1, 3/2 and 2 which ended up being 0, 3/4, 2, 15/4 and 6.
Lastly I found the area of the rectangles by multiplying 1/2 against the summation of the widths. My final answer was 25/4 which will be an overestimate because right endpoints on an increasing function will account for addition area above the curve.

For the function \(\displaystyle f(x)=x^2+x\) on the interval \(\displaystyle [0,2]\) the righthand Riemann sum:
the length of \(\displaystyle [0,2]\) is \(\displaystyle 2\) so that a four part regular partition is \(\displaystyle p_0=0,~p_1=0.5,~p_2=1,~p_3=1.5,~p_4=2\)
the righthand sum is \(\displaystyle \sum\limits_{k = 1}^4 {f({p_{_k}})({p_k} - {p_{k - 1}})} = (0.5)\sum\limits_{k = 1}^4 {f({p_{_k}})}\)