Evaluate the upper and lower sums for
f(x) = 2 + sin x,

0 ≤ x ≤ π,
with
n = 2, 4, and 8.

(round your answers to two decimal places.)

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: 
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 subinterval if that subinterval lies in [0, π/2], and at the right endpoint of the subinterval 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)). 

the lower sum for n=8 is 
(π/8)(f(0)+f(π/8)+f(π/4)+f(3π/8)+f(5π/8... 
and the upper sum is 
(π/8)(f(π/8)+f(π/4)+f(3π/8)+f(π/2)+f(π/...

Do the math. she originally had 50 dollars on her gift card and she had 12 dollars left on her gift card after she finished her shopping spree. if you subtract 50 minus 12 you get thirty eight dollars as your answer. so therefore, ciara spent 38 dollars total o