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.)

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

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The upper sum of the integral

`int_a^b f(t)dt`


`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:




`approx 8.18`




`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:


`approx 8.65`



`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.

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