Sigma notation with area under a curve, lower and upper rectangles

Question

I got the answer to this problem from my teacher but I don't understand why that's the answer:

y=√1+x^3 , domain 1≤x≤2 and with 50 intervals therefore width is 1/50. 

The Sigma notation for the lower rectangle is (1/50)∑√(1+(x/50)^3) with a 99 written above Σ and the what's written below Σ being x=50. Why isn't the upper limit n-1 aka 49 and the lower 0? Is my teacher wrong or am I misunderstanding sigma notation?

Answer 1

Using n from 0 to 49 the expression would look like this

 

(1/50)Σ√(1+(1+n/50)^3)

 

Then it could be written

 

(1/50)Σ√(1+(50/50+n/50)^3)     because 1=50/50

 

Or

 

(1/50)Σ√(1+((50+n)/50)^3)

 

If we put x=50+n then the limits will now be x=50+0=50 to 50+49=99

 

So we end up with

 

(1/50)Σ√(1+(x/50)^3) with x from 50 to 99.  The two expressions are equivalent.

 

Hope that helps.