What are the bounds of integration?

so the limits are 2 and 9

We want to express  as a integral. To do this, we have to identify  as a Riemann Sum that approximates the integral. (taking the limit makes the approximation equal to the value of the integral)

In general, to find a Riemann sum that approximates the integral of a function f over an interval [a,b] we can the interval in n subintervals of equal length and approximate the area (integral) with rectangles in each subinterval and them sum the areas. This is equal to

, where  is a selected point of the subinterval.

In particular, if we select the ending point of each subinterval as the , the Riemann sum is:

.

Now, let's identify this in  .

The integrand is x² so this is our function f. When k=n, the summand should be  because the last selected point is b. The last summand is  thus b=9 and b-a=7, then 9-a=7 which implies that a=2.

To verify our answer, note that if we substitute a=2, b=9 and f(x)=x² in the general Riemann Sum, we obtain the sum inside the limit as required.

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step-by-step explanation: