# Write the Riemann sum to find the area under the graph of the function #f(x) = x^2# from #x = 1# to #x = 5#?

# int_1^5 \ x^2 \ dx = 124/3 #

We are asked to evaluate:

Using Riemann sums. By definition of an integral, then

That is

And so:

Using the standard summation formula:

we have:

Using Calculus

If we use Calculus and our knowledge of integration to establish the answer, for comparison, we get:

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The Riemann sum to find the area under the graph of the function (f(x) = x^2) from (x = 1) to (x = 5) is given by:

[\sum_{i=1}^{n} f(x_i) \Delta x]

where (\Delta x) is the width of each subinterval and (x_i) is a sample point in the (i)th subinterval.

In this case, the interval ([1, 5]) is divided into (n) subintervals of equal width:

[\Delta x = \frac{5 - 1}{n} = \frac{4}{n}]

The sample points (x_i) are chosen to be the right endpoints of each subinterval. So, (x_i = 1 + i \Delta x) for (i = 1, 2, ..., n).

Substituting (f(x) = x^2) into the Riemann sum formula, we get:

[\sum_{i=1}^{n} (1 + i \Delta x)^2 \Delta x]

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