Evaluate the upper and lower sums for f(x) = 2 + sinx with n=2 I've got the upper as 7.85 but i cant figure out how to do the lower sum... upsum = (pi/2)(2 +sin(pi/2)) + (pi/2)(2 + sin(pi)) = 7.85 how do I do the lower? is the upper even right? thanks

Expert Answers

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It appears that you are trying to estimate the area under the curve y=2+sin(x) from 0 to `pi` :

(1)Upper sum with n=2: The width of the rectangles is `Delta x=pi/2` . The height of the rectangles is the maximum the function takes on the intervals -- in both cases the maximum occurs at `x=pi/2` .

So the upper sum is `(2+sin(pi/2))pi/2+(2+sin(pi/2))pi/2=3pi~~9.42`

(2) The lower sum with n=2 : again `Delta x=pi/2` . Now we take the lowest value the function takes on each interval; in both cases that is 2.

Lower sum is `(2+sin(0))pi/2+(2+sin(pi))pi/2=2pi~~6.28`

The average of the upper and lower sums is approximately 7.85.

The actual value is approximately 8.2831853. (The value is exactly `2+2pi` )

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