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Sums of Riemann | Exercises, first part

For the resolution of the exercises presented below, we have to apply the following Riemann sum:

\displaystyle \sum_{i=1}^{n} f(x_{i})\Delta x

Where \Delta x and x_{i} are calculated as follows:

\Delta x = \cfrac{b - a}{n}

x_{i} = a + i\Delta x

Maybe we will need some formulas of summations that you can find by clicking here.

Exercise 1 of the Riemann sums

Evaluate f(x)=x in the range [-3,1] using Riemann sum and then check the result using the corresponding definite integral.

The value of the interval that is more to the left is a=-3 and the other is b=1, let’s consider the equations of \Delta x and x_{i}:

\Delta x = \cfrac{1-(-3)}{n} = \cfrac{4}{n}

x_{i} = -3 + i \cfrac{4}{n} = -3 + 4 \cfrac{i}{n}

Now, let’s propose the sum to be solved, we will substitute \Delta x for what we calculate and we will substitute all x of the function f(x) for what we have calculated in x_{i} previously:

\displaystyle \sum_{i=1}^{n}f(x_i)\Delta x = \sum_{i=1}^{n}\left( -3 + 4 \cfrac{i}{n}\right) \left( \cfrac{4}{n} \right)

Perfect, now we will multiply the two parentheses we have in the summation:

\displaystyle \sum_{i=1}^{n}\left( \cfrac{-12}{n} + 16 \cfrac{i}{n^{2}}\right)

By properties of summations, our sum of the summation will be divided into a sum of summations:

\displaystyle \sum_{i=1}^{n}\cfrac{-12}{n} + \sum_{i=1}^{n}16 \cfrac{i}{n^{2}}

Everything that is not i can leave the summation by properties of the summations because everything that is not i is considered a constant:

\displaystyle -\cfrac{12}{n} \sum_{i=1}^{n} 1 + \cfrac{16}{n^{2}}\sum_{i=1}^{n} i

Now we will apply the sum of a constant and the sum of i that you can find in this article and then proceed to simplify:

-\cfrac{12}{n}n + \cfrac{16}{n^{2}} \left[ \cfrac{n(n+1)}{2} \right] = -12 + \cfrac{16}{n} \left[ \cfrac{n+1}{2} \right]

=-12 + 8 + \cfrac{8}{n} = -4 + \cfrac{8}{n}

Finally we apply the concept of limit, that knowing how it works, all the values that have a denominator n will be equal to zero and the values that do not have n are conserved:

\underset{n \to \infty} \lim \; -4 + \cfrac{8}{n} = -4 + 0 = - 4

Final answer: -4

Let’s check, if the integral of the function f(x) with the limits [-3,1] gives the same result, then our evaluation of the Riemann sum is correct:

\int_{-3}^{1}x \ dx = \left[ \cfrac{x^{2}}{2} \right]_{-3}^{1}

We evaluate:

= \cfrac{1}{2} - \cfrac{9}{2} = -4

Excellent, which means that our result of -4 is correct!

Exercise 2 of the Riemann sums

Evaluate f(x)=x^{2} - x in the range [1,2] using Riemann sum and then check the result using the corresponding definite integral.

Let’s calculate: \Delta x and x_{i}

\Delta x = \cfrac{2 - (1)}{n} = \cfrac{1}{n}

x_{i} = 1 + \cfrac{i}{n}

Now, let’s propose the sum to be solved, we will substitute \Delta x for what we calculated previously and we will substitute all the x of the function f(x) for what we have calculated in x_{i} previously:

\displaystyle \sum_{i=1}^{n}f(x_{i})\Delta x = \sum_{i=1}^{n} \left[\left( 1 + \cfrac{i}{n} \right)^{2} - \left(1 + \cfrac{i}{n} \right) \right] \left(\cfrac{1}{n} \right)

Let’s square the parentheses, remember that it is a perfect square binomial, therefore we will have the following:

\displaystyle \sum_{i=1}^{n} \left[ 1 + \cfrac{2i}{n} + \cfrac{i^{2}}{n^{2}} - 1 - \cfrac{i}{n} \right] \left( \cfrac{1}{n} \right)

Now let’s reduce terms of what is inside the brackets:

\displaystyle \sum_{i=1}^{n} \left[ \cfrac{i}{n} + \cfrac{i^{2}}{n^{2}} \right]\left(\cfrac{1}{n} \right)

We will multiply the bracket with the parenthesis:

\displaystyle \sum_{i=1}^{n} \left( \cfrac{i}{n^{2}} + \cfrac{i^{2}}{n^{3}} \right)

Then we will divide our sum of the sum in a sum of summations:

\displaystyle \sum_{i=1}^{n} \cfrac{i}{n^{2}} + \sum_{i=1}^{n} \cfrac{i^{2}}{n^{3}}

And everything that is not i can leave the sum because everything that is not i is considered a constant:

\displaystyle \cfrac{1}{n^{2}} \sum_{i=1}^{n} i + \cfrac{1}{n^{3}}\sum_{i=1}^{n} i^{2}

If we do not know the sum of i nor the sum of i^{2}, we can consult them by clicking here. After consulting them, it will be as follows:

\cfrac{1}{n^{2}} \left[ \cfrac{n(n+1)}{2} \right] + \cfrac{1}{n^{3}} \left[ \cfrac{n(n+1)(2n+1)}{6} \right]

For the moment, it only remains to simplify terms:

= \cfrac{1}{2n} [n +1] + \cfrac{1}{6n^{2}} [2n^{2} + n + n + 1] =

\cfrac{1}{2} + \cfrac{1}{2n} + \cfrac{1}{3} + \cfrac{1}{3n} + \cfrac{1}{6n^{2}}

Finally we will apply the concept of limit, where everything that has denominator n is equal to zero:

\underset{n \to \infty} \lim \; \left[ \cfrac{1}{2} + \cfrac{1}{2n} + \cfrac{1}{3} + \cfrac{1}{3n} + \cfrac{1}{6n^{2}} \right] = \cfrac{5}{6}

Final answer: \cfrac{5}{6}

Now let’s calculate the integral of the function f(x) in the range of [1,2]:

\int_{1}^{2}(x^{2} - 2) dx = \left[ \cfrac{x^{3}}{3} - \cfrac{x^{2}}{2}\right]_{1}^{2}

We evaluate:

= \cfrac{8}{3} - \cfrac{4}{2} - \cfrac{1}{3} + \cfrac{1}{2} = \cfrac{5}{6}

Excellent, which means that our result of \frac{5}{6} is correct!

Important notes:

  • If at the moment of arriving at the part of the evaluation of limits you realize that you have some n with positive exponent in some numerator, some calculation out there was not done well, then you have to review all the operations again.
  • It does not hurt to remember it (since sometimes it is forgotten), but remember that the sum of a constant is equal to n multiplied by the constant.
  • Courage, it’s easy to do these exercises!

Here you can visit the second part.

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