Use the definition to find an expression for the area under the graph of f as a limit.?

Question

This is the definition:
http://www.webassign.net/www28/symImages/2/5/3d2901ea25c4de1db1c01fa4fc587a.gif

This is the information given:
http://www.webassign.net/www28/symImages/5/0/511f2d8cdb5d33e354614d46e4fa37.gif

This is the answer:
http://www.webassign.net/www28/symImages/3/a/760fc431fd49351e81fe8fc0a03e85.gif

Why is it 1+10i/n (inside the radical) instead of just 10i/n?

Answer 1

The previous answer is completely incorrect. If you wanted to compute the rectangles with left endpoints instead of right, you would change the indices of summation from [i=1, n] to [i=0, n-1], but the formula inside the radical would remain 1 + 10i/n.

The reason why there is a 1 inside the radical is because you're integrating over the region 1≤x≤11, not 0≤x≤10. As such, you need to make sure the first point you evaluate the function is 1 and the last point you evaluate the function is at 11. And indeed, hen the formula inside the radical is evaluated with i=0, the result is 1, and when it is evaluated with i=n, the result is indeed 11. Were that 1 not there, the function would effectively be being integrated over the region [0, 10] instead of [1, 11].

Answer 2

Why is it 1+10i/n (inside the radical) instead of just 10i/n?

You can use 10i/n with i = 0 to i = n-1 in the summation.
Both wll give the same result.

If you use 1 + 10i/n, you are dividing the area into n larger rectangles, whereas when you use 10i/n, you are dividing the area into n smaller rectangles. As n increases and tends to ∞, Δx decreases and tends to zero. For any small finite value of Δx, sum of areas of small rectangles obtained by writing 10i/n is less than the area under the curve which is less than the sum of large rectangles obtained by writing 1 + 10i/n. But with limit n -> ∞, Δx -> 0 and in the limit all three areas become the same. So one can use either formula:
(i) writing 1 + 10i/n with summation i = 1 to i = n, OR
(ii) writing 10i/n with summation i = 0 to i = n-1.