Reimann Sum Limit Homework Solution | Integration Limits and Delta X Formula

[Qube]

Homework Statement

https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1419845_10201044047645089_1286462043_n.jpg?oh=adc74f67f112c0697cbfba79b4fa81fc&oe=5283F9AB

Homework Equations

delta x = (b-a)/n

The Attempt at a Solution

Well, from the delta x formula I can figure out the limits of integration. They're 4 and 0. That leaves us with three possible answer choices. I'm suspecting that the 4i/n term goes away and the answer is B, but I really don't know and I'm not even sure where to begin.

 

[Mark44]

Homework Statement

https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1419845_10201044047645089_1286462043_n.jpg?oh=adc74f67f112c0697cbfba79b4fa81fc&oe=5283F9AB

Homework Equations

delta x = (b-a)/n

The Attempt at a Solution

Well, from the delta x formula I can figure out the limits of integration. They're 4 and 0. That leaves us with three possible answer choices. I'm suspecting that the 4i/n term goes away and the answer is B, but I really don't know and I'm not even sure where to begin.

You have the interval [0, 4] that you will divide into n subintervals of equal length. How would you write x_i, the x value in the i-th subinterval? The x value could be at the left or right end of a given subinterval, or somewhere in the middle of it.

 

[Qube]

I'm not sure what terms to write xi in terms of. I guess, (x/n) would give me the width of each subinterval and I'm not sure what else.

 

[Mark44]

No, the width of each subinterval would be 4/n. Since the summation has cos(2 + ...), that's going to show up in the integral as well.

 

[Qube]

So the integral would have cos (2+x) as the integrand?

 

[Mark44]

Yes. Do you see how it works? Since i is running from 1 to n, 4i/n represents the x value at the right side of each subinterval, and cos(2 + 4i/n) is the function value associated with that x value.

 

[Qube]

Alright, I see :)! 4/n is the width of each sub interval. The i represents each sub interval.