Okay, let's evaluate the upper and lower sums for this function with n = 8 intervals:
1) Find the interval size: = /n = /8 =
2) Evaluate the function at the endpoints of 8 intervals:
f(0) = 2 + sin(0) = 2
f() = 2 + sin() = 3
f(/8) = 2 + sin(/8)
f(2/8) = 2 + sin(2/8)
f(3/8) = 2 + sin(3/8)
f(4/8) = 2 + sin(4/8)
f(5/8) = 2 + sin(5/8)
f(6/8) = 2 + sin(6/8)
f(7/8) = 2 + sin(7/8)
3) Upper sum:
U = 2 + (2 + 3)/2 + (2 + 2 + sin(2/8))/2 + (2 + 2 + sin(3/8) + sin(4/8))/2 + (2 + 2 + sin(5/8) + sin(6/8) + sin(7/8))/2
= 14 + 1.79 + 2.5 + 3 + 3.5 = 24.79
4) Lower sum:
L = 2 + (2 + 2)/2 + (2 + 2 + 2)/2 + (2 + 2 + 2 + 2)/2 + (2 + 2 + 2 + 2 + 3)/2
= 14 + 2 + 2 + 2 + 4 = 24
So the upper sum is 24.79 and the lower sum is 24.
Let me know if you need more details!