Question

Evaluate the upper and lower sums for f(x) = 2 + sin x? Evaluate the upper and lower sums for f(x) = 2 + sin x, 0 ≤ x ≤ π, with n = 2, 4, and 8.

Please explain clearly step by step with the answer.
What my answers were, were wrong. For instance for n=2 I did, pi/2 is my change in x. So what I did for my lower sum was (pi/2)(f(0)+f(pi/2)). And that was...

Answer 1

For the answer to the question above, For n = 2:
The smallest value of f(x) on [0, π/2] is 2, which occurs at x = 0. The smallest value of f(x) on [π/2, π] is also 0, which occurs at x = π. So the lower sum is (π/2)(2 + 2) = 2π

The largest value of f(x) on [0, π/2] is 3, which occurs at x = π/2. This is also true for the interval [π/2, π]. So the upper sum is (π/2)(3 + 3) = 3π

n = 4:

Note that f '(x) = cos(x), which is positive for [0, π/2) and negative for (π/2, π]. This tells us that f is an increasing function on [0, π/2) and a decreasing function on (π/2, π]. So for the lower sum you will always evaluate f at the left endpoint of the sub-interval if that subinterval lies in [0, π/2], and at the right endpoint of the sub-interval if it lies in [π/2, π]

Thus, the lower sum for n = 4 is

(π/4)(f(0) + f(π/4) + f(3π/4) + f(π))

and the upper sum is

(π/4)(f(π/4) + f(π/2) + f(π/2) + f(3π/4)).

I have no doubt that you will now be able to do the n = 8 case on your own.