Approximations questions

[dagreat45]

[h=1]Calculate the limit of the right-endpoint approximations for the area under the graph of f(x) on the interval?[/h]Calculate the limit of the right-endpoint approximations for the area under the graph of f(x) on the interval [1,4] as the number of subdivisions goes to infinity. Confirm this area geometrically.

f(x)=62x+124

lim(N->inf) RN=


Do you just find the derivative or what

 

[MarkFL]

I would first calculate the area of an arbitrary rectangle:

\(\displaystyle A_k=\left(\dfrac{x_n-x_0}{n} \right)f\left(x_k \right)\)

Using the given endpoints and function, we find:

\(\displaystyle A_k=\dfrac{3(62x_k+124)}{n}=\dfrac{186}{n}(x_k+2)\)

Now, we have that \(\displaystyle x_k=\dfrac{3k+n}{n}\) hence:

\(\displaystyle A_k=\dfrac{186}{n}\left(\dfrac{3k+n}{n}+2 \right)=\dfrac{558}{n^2}(k+n)\)

And so the total area is:

\(\displaystyle A=\lim_{n\to\infty}\left(\dfrac{558}{n^2}\sum_{k=1}^n(k+n) \right)\)

Can you proceed from here to find the limit?

For the second part of the problem, what shape is the area we are finding?