How do you evaluate the definite integral by the limit definition given #int x^2+1dx# from [1,2]?
1 Answer
Please see the explanation section below.
Explanation:
Here is a limit definition of the definite integral.
.
Where, for each positive integer
And for
I prefer to do this type of problem one small step at a time.
Find
For each
Find
And
Find
# = 2+(2i)/n+i^2/n^2#
Find and simplify
# = sum_(i=1)^n( 2/n+(2i)/n^2+i^2/n^3)#
# = 2/nsum_(i=1)^n (1) +2/n^2sum_(i=1)^n (i)+1/n^3 sum_(i=1)^n (i^2)#
Evaluate the sums
# = 2/n(n)+2/n^2((n(n+1))/2) + 1/n ((n(n+1)(2n+1))/6)#
(We used a summation formula for the sums in the previous step.)
Rewrite before finding the limit
# = 2+ ((n(n+1))/n^2) + 1/6 ((n(n+1)(2n+1))/n^3) #
Now we need to evaluate the limit as
and
To finish the calculation, we have
# = 2+(1)+1/6(2) = 10/3#