Questions regarding the Sigma notation

Question

I have a problem about estimating the area under a curve $f(x)$ with right endpoints using $4$ rectangles.

I know how to work through the mechanics of the problem but I am really trying to understand the underlying fundamentals better.

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If I use Sigma notation to write this problem out, my confusion lies in how the variable $i$ relates to $n$. $$ \sum_{i}^n f(x) $$ on simple problems usually $i = 1$ which leads me to believe that $i$ iterates through the sum of: $$ x_{1} + x_{2} + x_{3}...x_{n} $$

However, when I have a problem like: $$ f(x) = \cos(x) $$ which asks me to estimate its area with $4$ rectangles using right endpoints from $ [0,\pi/2]. $

I ultimately get lost in how to properly display this using sigma notation since in this case: $ n=4, $ which leads me to believe I should represent this as such: $$ \sum_{i}^{n=4} \cos(x) $$

BUT, that doesn't make sense according to a description of the sigma notation I read in my textbook which says: $$ \sum_{i = m}^n f(x_{i})\Delta x $$

which, according to the definition in my text, tells us to end with: $$ i=n $$

and to start with: $$ i=m $$

which means that if I have to find an Area under the curve between: $$ [0,\pi/2] $$

then it makes more sense to me, that I should really be thinking about this notation as follows: $$ \lim_{n\to 4}\sum_{i=0}^{n=\pi/2} cos(x)\Delta x $$

I say this because the nature of the problem requires that I break up this function to as close to 4 rectangle units as possible depending on whether I need the left or right end point (or midpoint). The iteration starts at i = 0, and terminates at $$\pi/2$$ as needed...

Is this the right way to think about this problem and even more important am I using this notation correctly in the manner that I have described?

Answer 1

First of all, $\sum_{i=1}^nf(x)$ does not make sense because it literally gives $$ \sum_{i=1}^nf(x)=n\cdot f(x). $$

Also, it is completely nonsense (no offense, I'm just talking about the notations here) to write $$ \lim_{n\to 4}\sum_{i=0}^{n=\pi/2} cos(x)\Delta x. $$ Because

• $n$ is a positive integer, it does not make sense to write $n=\pi/2$.
• There is no limiting process here.
• You should have $x_i$ in $\cos(x)$ instead.

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Suppose $ f(x) = \cos(x) $. Estimate its area under $f$ with $4$ rectangles using right endpoints from $ [0,\pi/2]$.

The sum you are looking for is something like $$ \sum_{i=1}^4f(x_i)\Delta x. $$ What you need to figure out is $\Delta x$ and the $x_i$'s. There is no limiting process here.

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Take a careful look at this example: