Show all work and reasoning

which of the following limits is equal to ?

a.

b.

c.

d.

Split up the interval [2, 5] into equally spaced subintervals, then consider the value of at the right endpoint of each subinterval.

The length of the interval is , so the length of each subinterval would be . This means the first rectangle's height would be taken to be when , so that the height is , and its base would have length . So the area under over the first subinterval is .

Continuing in this fashion, the area under over the th subinterval is approximated by , and so the Riemann approximation to the definite integral is



and its value is given exactly by taking . So the answer is D (and the value of the integral is exactly 39).