Question

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

Accepted Answer

The upper sum of the integral
int_a^b f(t)dt 
is
U=sum_{i=0}^{n-1} f(x^+) Delta x  where Delta x = {b-a}/n , f(x^+)=max(f) on the interval [x_i,x_{i+1}]  and x_i=a+i Delta x 
The lower sum is 
L=sum_{i=0}^{n-1} f(x^-) Delta x where f(x^-)=min(f)  on the same interval
Since f(x)=2+sin x , and the bounds on the integral are a=0 and b=pi , then  
Delta x=pi/n 
x_i={i pi}/n 
For n=4, this means we have:
U_4=(f(pi/4)+f(pi/2)+f(pi/2)+f({3pi}/4))pi/4 
=pi/4(8+2sqrt2/2+1) 
=pi/4(9+sqrt2) 
approx 8.18 
L_4=(f(0)+f(pi/4)+f({3pi}/4)+f(pi))pi/4 
=pi/4(4+2sqrt2/2) 
=pi/4(4+sqrt2) 
approx 3.39 
Noticing the symmetry of the function, we see that we really only need to double the sum up to x=pi/2 .
When n=8, we should have more refined values and get after the doubling:
U_8=pi/4(f(pi/8)+f(pi/4)+f({3pi}/8)+f(pi/2)) 
approx 8.65 
Similarly, 
L_8=pi/4(f(0)+f(pi/8)+f(pi/4)+f({3pi}/8)) 
approx 6.29 
The upper and lower bounds for n=4 are 8.18, 3.39.  The upper and lower bounds for n=8 are 8.65, 6.29.

Math

Evaluate The Upper And Lower Sums For F(x) = 2 + Sin X, 0 ≤ X ≤ π, With N = 2, 4, And 8. Illustrate With Diagrams Like The Figure Shown Below. (round Your Answers To Two Decimal Places.)

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