Answer:
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.
